 So a very good morning to all the students. So in conic sections, we have already discussed all the exercises of the circles. So today we will start a new chapter in conic section that is parabolic. So without wasting any time, let's move to the first exercise this exercise one. So here's our first question. It is in the vertex of the parabola. So one parabola is given here and we have to identify its vertex. Means we have to find the coordinates of its vertex. Now parabola, what is the parabola given? It is given as y squared, y squared plus 6x minus 2y plus 13 equal to 0. Okay. So what we will do, we will try to make a perfect square in y and then we will compare with our standard form of the parabola. So let's do it. This will be y squared minus 2y plus 1. Let me add plus 1 and subtract 1 from this. So actually it will become our y minus 1 whole square. So now this will be the rest of the things will remain as it is. So plus 6x plus 13 equal to 0. So it will become y minus 1 whole square. Then take these things to the right hand side, it will be minus 6 plus 12 means minus 12. So further if you see it can be written as if we take minus 6 common, it will be x plus 2. So this parabola is of the form this, our standard parabola minus 4x. So we will compare with this. So what is this? Our equation is basically a equation of parabola with the shifted vertex. So what will be vertex for this parabola? It will be basically, if you see vertex for this, what we used to get vertex, this y equal to 0 and our x is equal to 0. So similarly if you compare it here, we can get, we will get this y minus 1 equal to 0 and our x plus 2 or x plus 2 will be equal to 0. From here we get x is equals to minus 2. And from here we get y equal to 1. So basically it will be, how this parabola will look like? Let me draw rough sketch. So it will be basically like this way. So its vertex will be here. Its vertex will be, the coordinates of vertex will be minus 2 comma 1, minus 2 comma 1. And its axis, its axis will be parallel to our x axis. Its axis will be parallel to our x axis. What does it mean? This will be our y coordinate, but this will be our y axis and our x axis will be something like this. So this parabola represents, is of the form y is equal to minus 4x with shifted vertex. So vertex is minus 2 comma 1. So basically this option A is given here, minus 2 comma 1. So this will be the vertex of this parabola. So let's move to the next question. This question number 2. It is given if the parabola y square is equal to 4x, 4x passes through 3 comma 2, then the length of the lattice vector. So the standard form of the parabola is given here. So no issue at all. This is our y square is equal to 4x. Now this parabola is passing through this 3 comma 2. This parabola passes through any point p. Suppose I am taking this point as p, whose coordinates are 3 comma 2. So this point must satisfy the equation of parabola. So it will be equal to 4 is equal to 4A into 3, right? 4A into 3. Now for this parabola, what is the length of the lattice vector? What is the length of lattice vector for this standard parabola? The length of lattice vector is actually 4A. So we actually need to find the value of this 4A, right? So from here, from this equation you see 4 by 3 will be equal to 4A. So this will be our answer. This will be the length of lattice vector, right? This will be the length of lattice vector. So this option is there. This is what is asked in the question, the length of lattice vector. So it will be simply 4 by 3. Now let's see the next question, question number 3. It is saying the value of p such that y equal to x square plus 2px plus 13 is 4 units above the x axis. Okay, so if you observe, the equation of parabola is given as y is equal to x square plus 2px plus 13, right? So basically this is of the form, this ax square plus px plus c. This parabola is of this form, right? And here a is greater than 0, like a is equal to 1, that is greater than 0. So what does this mean? This mean this will be this parabola. This parabola represents upward opening parabola. It represents upward opening parabola. It represents upward opening parabola since a is equal to, sorry, a is equal to 1, that is greater than 0. So it represents upward opening parabola. And its axis will be, its axis will be parallel to, its axis will be parallel to y axis, right? So it will be parallel to y axis. Okay, so if we draw a rough sketch like this will be on rough sketch of this parabola, right? So this will be the axis of the parabola, okay? And there will be x axis somewhere around here. So this will be on x axis, okay? And this is our vertex, okay? Its distance from x axis is, this distance is given as 4 units, right? This distance is given as 4 units. Now for this type of parabola, for this type of parabola, parabola of this form, the coordinates of VR, the coordinates of vertex, coordinates of vertex is minus V upon 2n comma minus V upon 4n. This will be the coordinates of vertex of parabola of this form. So if you observe here, this minus D upon 4a, what will be the value of D basically in this equation in this particular given equation? It will be D squared means 4p squared minus 4a into c, that is 13. The y coordinate, the y coordinate of vertex, y coordinate of vertex will be equal to minus D upon 4a. That is nothing but minus D means we can write it as 4 into 13 minus 4p squared upon 4a, that 4a is 1. So this will be 1. And this y coordinate is nothing but the distance from the x axis, right? So it will be equal to 4. So from here if you see, this 4, 4 will be, gets cancelled out, means we can simply cancel it off. So it will become 13 minus p square is equal to 4. From here we get p square is equal to 9 or p is nothing but plus minus 3, okay? So this will be the value of p, right? This will be the value of p for the vertex of this parabola to be 4 units from the x axis. Let me write here also, the coordinates of this v will be minus b upon 2a. This will be the x coordinate and the y coordinate will be equal to minus D upon 4a, okay? So hope this is clear to all. So let's check this next question. So in the question number 4, it is given the length of the lattice rectum of the parabola whose focus is 3, 3 and directrix is 3x minus 4y minus 2. Okay, so let me draw one rough sketch for this. So we are drawing parabola in this paper, there is no other options like anyhow. So this is our parabola, let me erase some part of it, okay? And then directrix is also given and focus is also given for this parabola, okay? So here if you see the focus of this parabola is, let me denote it as s, okay? The focus is given as 3, 3. And the equation of directrix, this equation of directrix is given as 3x minus 4y minus 2 equal to 0, okay? Now we need to find the length of lattice rectum. So in parabola, this vertex, this is our vertex, no? So actually vertex, let me call this point as p, the intersection point of this axis and this is our axis of parabola. So let me call this intersection point of p, sorry, axis and directrix as p. So basically this v vertex, vertex is the midpoint, is the midpoint of what p is? Means vertex v, which I am representing as v, so v is the midpoint of p is. And what is this distance? If you observe, this distance is basically a and this distance is also a. Hope these things are clear to everyone, like while having studied this parabola chapter, you came across all these questions and hope this is known to all of you, right? So what I will be doing here, I will try to find the value of a. So how can we find, like we can find the perpendicular distance from s on this line, like we can find this sp, no? Sp, we can find it out. What will be that? It will be perpendicular distance from focus on this line. So let me write it, this will be 3 into 3 minus 4 into 3 minus 2 in mod and under root of a square plus b square, that is nothing but 3 square plus 4 square and this is equal to 2 times a. Okay. So how much it's coming? If you see this will be 9 minus 12 minus 14. So minus 14 plus 9 that is minus 5. Minus 5 come out upon this will be 5 is equals to 2. So basically our 2 a comes out to be 1. From here we get a is equal to what? 1 by 2, right? And what we are going to find in this question? Okay, the length of lattice rectum. So length of lattice rectum is nothing but length of lattice rectum is nothing but 4 times a, that will be 4 times of this half that will be equal to 2 units, right? So this option b is correct. Now let's move to the next question, question number 5. Okay. So here it is. If the vertex and focus of parabola are these and this respectively then it's a question. Okay. So we are provided with the vertex and focus of a parabola and we need to find the equation of parabola. Okay. So what is the vertex? The coordinates of vertex. The coordinates of vertex are this 3 comma 3, right? And coordinates of focus are minus 3 comma 3. Okay. So if you observe the Y coordinate of this vertex and focus is same. This Y coordinates are same. Y coordinates are same. So one thing is sure, like our parabola, our parabola's axis will be what you say parallel to x axis. This is clear from this coordinates, coordinates of this vertex and focus. Okay. And so what can be our parabola means how our parabola will look like? So suppose this is our x axis. Okay. So this is our suppose x axis. So the axis of parabola will be parallel to this. Okay. So our parabola will look something like this. So this will be our vertex and this will be our focus. Okay. So what is the coordinate of focus? This is minus 3 comma 3 and coordinates of vertex is 3 comma 3. Okay. Coordinates of focus is minus 3 while x coordinate of vertex is 3. So not this one. Our parabola will actually look like this. Our parabola will be like this. This will be the, this is our x axis. This will be the axis of parabola. Its vertex is given as what? 3 comma 3 and its focus is minus 3 comma 3. Okay. So basically our parabola will be of this form. Right. So this will be basically of what you say. How can we write the standard parabola of this form? We write it as y square is equal to minus 4x. Right. y square is equal to minus 4x. Now we know this is the case of the shifted vertex. Right. This is the case of the shifted vertex. So our equation will be like case of shifted vertex. So our equation of parabola will be this. y squared means y minus vertex square. Right. So y minus alpha squared plus sorry. No plus. So y minus 3 squared is equal to minus 4a. x minus 3. Right. We normally used to write in this form. No. y minus alpha whole squared is equal to minus 4a into x minus beta. So what is alpha here? Alpha is 3 here. What is beta here? Beta is also 3. Vertex coordinates of vertex. Right. And if you see this 4a, this 4a thing, not 4a, this distance will be basically a. This distance will be a. Right. So if you see our a will be, our a will be equal to what? 3 minus of minus 3. That will be equal to 6. Okay. Let me write it as capital since I'm taking capital. So now open it. So it will be y squared plus 9 minus 6. Right. Is equal to minus 4 into a that is minus 24x minus 3. So it will be basically y squared minus 6 y plus 24x plus 24x. This will be plus 72 and while coming to this side, it will be minus 72 plus 9. That is minus 6 to 3 is equal to 0. So this will be our, this will be our final answer. This will be the equation of the required 7 y square minus 6 y plus 24x minus 6 to 3 equal to 6. So this option C is correct. Okay. So let's take the next question. This question number 6. Okay. So given here, if the vertex of the parabola y is equal to x square minus 8x plus C lies on x axis, then the value of C. Okay. We have done this similar type of question earlier also. I think second or third one. So this is of the form a square plus bx plus 6. Right. This parabola is of this form. So this is that y is equal to x square minus 8x plus C. Okay. Its vertex is lying on the x axis. Its vertex is lying on x axis. Okay. So this is our x axis. Okay. And this is our vertex. This is vertex. And then it's a y coordinate. It's y coordinate of vertex. Vertex, y coordinate, y coordinate, y coordinate of vertex will be equal to 0. So what is the y coordinate of vertex? It is minus d upon 4a minus d upon 4a is equal to 0. Now, what is the value of this d? This is b square. B square means what? 64 minus 4 times a is 1 and C. So this will be minus of 64 minus 4c upon 4a. 4 is equal to 0. So from here, if you see minus 64 plus 4c is equal to 0. So from here, we get C is equal to 60. C equal to 60. So this will be our answer. Option C is correct. Now, see this question number seven. It is saying the parabola having its focus at 3, 2 and directrix along the y axis as its vertex at. Okay. So let me draw one sketch for this. So parabola is given and its vertex is to us. Okay. So this one is parabola. Directrix is along the y axis. Okay. Its direct axis is along the y axis. So let me call it as directrix and it will be along y axis. Let me draw the axis of the parabola also. And what is known to us? Okay. Its focus is known to us. We need to find the vertex. Okay. I think we are done with this. So this is our focus. Okay. This is our focus is whose coordinates are 3, 2. And this is our directrix whose, which is along the y axis. This means this is x equal to 0 is the basically the directrix of this parabola. Directrix of this parabola. So we need to find since x equal to 0 means this is y axis particularly. So if you observe, if this x equal to 0 is the directrix of this parabola, means this axis will, axis of this parabola will be parallel to this axis of parabola will be parallel to x axis. Hope you all will agree with this. So axis of parabola will be parallel to x axis. Why? Because this axis of parabola and this directrix are mutually perpendicular to each other. Means perpendicular to each other. Now we need to find this vertex. We need to find the coordinates of vertex. So let me call this point of interaction of intersection of this axis and directrix as P. So we will be the midpoint, right? So we will be, we will be midpoint of this piece. Midpoint of piece, right? So how can we find and what will be the coordinate of this P? What will be the coordinate of P? Its coordinate will be this, its x coordinate will be 0 because it will lie on the directrix. And what will be its y coordinate? Its y coordinate will be equal to y coordinate of focus. So why? Because its axis is parallel to x axis. So we know the coordinates of P. We know the coordinates of S. We can easily find the coordinates of V. So what would be the coordinates of V? It will be this 3 plus 0 upon 2 comma 2 plus 2, 2 plus 2 upon 2. This will be the coordinate of V. That will be nothing but 3 by 2 comma 2, right? 4 divided by 2 will be 2 only. So any options given here? 3 by 2 comma 2, yeah, seventh, this option B is correct. So let's move to the next question. Discussion number 8. The directrix of the parabola, this, okay? We need to find the directrix of this parabola. So as usual, we will try to make a rough sketch for this, okay? We will try to make a rough sketch for this. There is no option for this shape of parabola. So we have to draw this ellipse and circle and then we have to erase it. Okay, anyhow, no. This parabola we have drawn and we need to find this equation of this. It's directrix, okay? So let's try this out. Equation of parabola is given to us, right? So this equation of parabola is basically x square minus 4x minus 8y plus 12 equal to 0. Let's try to make a perfect square in x. Then we will compare with the standard form and we will have the coordinates of vertex, right? So how can we make it a perfect square in x? So x square minus 4x means x square minus 2 into 2x. That is, I'm adding plus 4 and I'm subtracting 4, okay? So it will be basically x minus 2 whole square. So plus 4 minus 4 we have done. What are the rest of the things that will be remain same? That will remain same. That will minus 4x and this plus 12 equal to 0. So this will become x minus 2 whole square, right? So x square minus 4x plus 4, okay? So it's fine. And take these things to the right-hand side. It will be 4x plus 12 minus 4 plus 8. Well, coming to this side, it will be minus 8, right? Minus 4x will become 4x. This thing will become minus 8, okay? I'm taking 4x again here. It will be minus 8 y, no? 8 will be minus 8 y, okay? So this will be 8 y. This will be 8 y minus 8, okay? So I think it's clear. So x minus 2 whole square is equal to 8 times y minus 1. Now what we can do? It's a case of the shifted vertex. And the standard form for this type of parabola will be x square is equal to this 4a y. This will be the standard parabola of this form. So what will be vertex for this? Vertex for this will be x minus 2 equal to 0, x minus 2 equal to 0. From here we get x is equal to 2. And this y minus 1 equal to 0 or y equal to 1. So our vertex having the coordinates of this 2 and 1. So the coordinates of vertex we got and the coordinates of our focus. Okay, that's not given. We are having the, we have to find the equation of parabola. Okay, we will try it out. Now if you observe, we can, what else can we find from here? We can find the value of a also. So like 4a is equal to 8, means our a is equal to 2. Okay, so these three informations we found the, this vertex coordinates, coordinates of vertex and we found the value of this a. Now we need to find the directrix of parabola. So this is our directrix basically. This is our directrix. So for this type of, this form of parabola, our directrix is basically this y equal to minus 8. For this, directrix is equal to, directrix is equal to what will be directrix? It x equal to, it is of the form x square is equal to 4a y, you know. Okay, so basically this parabola will be not like this. This is wrong basically. Our parabola will be upward opening. Upward opening. So our parabola will be not of this form. It will be of this form. So this is wrong sketch actually. Why? Because it is of the form x square is equal to 4a y. So it will be, our parabola will be like this. Okay, our parabola will be like this with shifted vertex. And our directrix will be somewhere here. So let me take it as directrix. This will be our directrix. This will be our directrix. And what is the coordinates of this vertex? Coordinates of vertex is coming out to be 2 comma 1. Okay. So what will be a directrix for this standard parabola? It is basically y equal to, y equal to a, right? Y equal to a, y equal to a or y minus a equal to 0, right? So what is y in this case? It is y minus 1, y minus 1, sorry. For this standard parabola, this will be y equal to minus a. So we can say this y plus a, y plus a is equals to 0. This is the directrix for this standard parabola. So let me write it clearly. It will confuse you all. So it will be y plus a equal to 0, directrix for this standard form. So for this form, it will be y minus 1. Like in place of y, we will write y minus 1. The value of a, that is 2, this equal to 0. From here, we get y equal to y plus 1 equal to 0 or y equal to minus 1. So this will be the directrix of our required circle. Okay. So, okay. Let me represent it by capital Y. Otherwise, it will be a problem. Since we have taken the standard form as this, no. So it's directrix will be y plus a equal to 0. Now replace the value of y in our original parabola. It's basically y equal to y minus 1. So y minus 1 plus a, we have found it out already. So y minus 1 plus 2 equal to 0. From here, we get y equal to minus 1. That will be our answer to this question. So this option C is correct. Okay. Now, this parabola, what we have drawn earlier, that is wrong basically. This is the right parabola. So that's why after writing the equation, we could have like, we could have drawn the parabola. But earlier before writing the equation, we have drawn. So it came out to be wrong. So this question number nine, let's check it out. The equation of lattice rectum of parabola is this. Okay. So one parabola is given here. This x square plus 4x plus 2y equal to 0. Okay. So we need to find the equation of lattice rectum. So first, let me try to write in the form of standard form of parabola. So I will try to make the perfect square in x. So this will be x square plus 4x. What can we do? We can add plus 4 and we can subtract 4 from this. So what it will become? It will become x plus 2 whole square x square plus 4 plus 4x, right? And take it to the right-hand side. It will be minus 2y plus 4. Okay. So x plus 2 whole square take minus 2 common. This will be y minus 2. Okay. So it is of this form. So x square is equal to minus 4a y with shifted vertex, no doubt. Right. So we can now make a rough sketch. Okay. We don't need a sketch for this also. So from here, if you see, while comparing this, while comparing this, what we got? We got 4a. The value of 4a is equal to 2. So a will be equal to basically. A will be equal to 1 by 2. Okay. Okay. We need, I think we need a sketch for this. So let me draw, let me draw the one rough sketch for this. So this will be basically, yeah, it will be like downward opening parabola, right? And we need to find the equation of lattice rectangle. So let me draw the axis also. X is for this parabola. It will be our axis. It will be our direct trace. Okay. And let me draw this lattice rectum also. Okay. Okay. So if you see, we need to find this equation of this. Lattice rectum, right? And this is our vertex. Okay. So what will be vertex for this basically? The vertex for this will be, vertex for our parabola will be minus 2. Like what we will do? We will equate this x plus 2 equal to 0. From here, we get x equal to minus 2, right? And we will equate this y minus 2 equal to 0. From here, we get y equal to 2. This is what we get. So the vertex will be basically minus 2 upon 2. So this vertex will be minus 2 upon 2. And A is coming out to be half, right? So this distance is basically A, no? This is our A. This is our A. So the equation of AB if you see, equation of AB will be, this will be minus half. And what will be this focus if you see? What will be this focus? So basically, let me write the coordinates of focus. So it will be minus 2, sorry, it will be minus 2 and minus of this A. This is A and since it is a opening downward parabola, downward opening parabola. So it will subtract like the y coordinate, the x coordinate of this, okay. The x coordinate will remain same. The x coordinate of this will remain same since it is a downward opening parabola. Its axis will be, its axis will be basically, axis of this parabola, axis of this parabola. Will be parallel to, will be parallel to y axis, right? So that's this, what do you say? The x coordinate of focus, the x coordinate of focus will be same as the x coordinate of, x coordinate of focus will be same as the x coordinate of vertex. That is nothing but minus 2. And what will be our y coordinate of focus? Y coordinate of focus, it will be reduced by this half, means by this A. So our y coordinate will be 2 minus this A. That is nothing but minus of half. So 2 minus half is 3 by 2, okay. So this is the, what do you say? This is what we got, the coordinates of S. But whether we need to find that, we need to find the equation of lattice vector, right? So this lattice vector, if you see, what will be slope of this? What will be slope of AB? Slope of AB will be equal to 0 only. Slope of AB will be equal to 0 since it is parallel to x axis, right? So y, and it is passing through, AB is passing through this focus, okay? So it must satisfy, so y equal to this 3 by 2. This will be our equation of AB, right? Y equal to 3 by 2. This will be our equation of AB because this AB is parallel to our x axis, okay? So we could have directly found the y coordinate of focus only and we could have written the equation of AB. Anyhow, so on simplify, if you see, this will be nothing but 2y equal to 3 or 2y minus 3 equal to same thing. So option C is given here. It will be our equation of lattice vector. AB is our lattice vector. So we are done with this question number 9. Now the focus of the parabola. So one parabola is given here, x square, x square minus 8x plus 2y plus 7 is equal to 0, okay? And we need to find the focus of the parabola. So let me write it as x square minus 8x means 2 into 4. I think we have to add and subtract 16. So this will become x minus 4, x minus 4 whole square, right? Plus 2y plus 7 minus 16 is equal to 0. So it will become x minus 4 whole square, x square plus 16 minus 8x, right? And what this thing will become, it will be minus 2y and minus 16 plus 7. So minus 9, while coming this side it will be plus 9, okay? So this will, further we can write it as x minus 4 whole square minus 2, take minus 2 common because we have to represent it in the standard form. So it will be y and minus 9 by 2. Is it okay? Minus 2y plus 9, the same thing. But why we have taken this minus 2 common to make it as in the form of our standard parabola, that is nothing but our x square is equal to minus 4a by 9. So what will be the vertex of this vertex? Vertex of this parabola will be basically minus 4, sorry, plus 4, means x minus 4 we will equate this to 0. So from here we get x equal to 4 and y coordinate will be 9 by 2, right? And it will be x square is equal to minus something, right? So it will be a downward opening parameter. It will be a downward opening parameter. So a rough sketch if you say the rough sketch of this parameter will be something like this. And so we got this, we got the value means we got the coordinates of vertex, but we need to find the focus, right? So vertex is basically 4 comma 9 by 2. Okay, we can do one thing. From here we can find the value of A also. So this 4a is equal to 2. So what will be A? A will be equal to 1 by 2, okay? So since this is a downward opening parameter, right? So the y coordinate will be, x coordinate of focus will be same. X coordinate of focus will be same, but this y coordinate will be, let me write it, x coordinate of focus will be equal to x coordinate of vertex that is equal to 4. And y coordinate of focus will be equal to, it will reduce by this A, A quantitation. Since this is our A, this distance is basically A. So it will be 9 by 2 minus half, right? So 9 by 2 minus half means what? This 4.5 means 4. It will also become 4. So this will be our coordinate of focus, this 4 comma 4, okay? So this option B is correct. So we are done with this. Let's move to the next question. Question number 11, okay? The equation of parabola with focus, this and directrix, this. So x equal to minus 3 is the directrix of parabola and focus. Okay, so this is our directrix, okay? x is equal to minus 3. So its axis will be, the axis of parabola will definitely be parallel to x axis. And we are given the focus also, and we need to find the equation of parabola, okay? So this is our directrix basically, directrix of parabola. And its equation is x plus 3 equal to 0, means x equal to minus 3, okay? And this is our focus, this is our focus which coordinates are 3 comma 0, 3 comma 0, okay? So definitely if you see the parabola will be of the form y square is equal to, y square is equal to 4ax. Our parabola, our required parabola will be of this form, right? And if you see what is, what will be the value of a here? What will be the value of a here? Now first let me find the equation of, sorry, the coordinates of vertex. Let me find first the coordinate of vertex. So its y coordinate will be 0 only, its y coordinate will be 0. And if you see this two way, this is a, right? This whole distance between this focus and directrix is 2. So this 2a is equal to, this 2a will be equal to how much? 3 minus 3, that is 6. So a will be equal to 3 basically. So a will be equal to 3. So basically the vertex will lie on 0 comma 0. This will be the vertex, right? So this will be our vertex. So the form of our equation will be, equation of parabola will be equal to y equal to, y square is equal to 4ax, right? So since the vertex of this is coming out to be, what is it? So what is the value of a? So 4 into 3 will be 12, 12x, y square is equal to 12x. So this will be our equation of the required parabola. Okay, you could have obtained the vertex by using that method also, like we will be the midpoint of this piece, right? From there, you can calculate the, sorry, you can get the coordinates of vertex. And from the concept of this a, also we got the equation, sorry, the coordinates of vertex. So actually this will be our, if you say no, this will be our y axis. So basically it becomes our standard parabola. So this will be our y axis and its axis will be the axis of parabola will be nothing but x axis, right? So this was our question number 11. Let's take it, the next question, just a minute. So yeah, take, now we have to take question number 12. So what is this question saying? It is in equation of the parabola whose axis is parallel to y axis and which passes through the points this, this and this. Okay, so basically we are having one parabola whose axis is parallel to y axis. So let me draw one rough sketch of the parabola. So this will be like rough sketch of our parabola whose axis is parallel to y axis, right? So it will be basically a upward opening parabola, okay? So for such parabolas, we can assume the equation as y is equal to this A axis square, right? So this is A plus Bx plus C, okay? And it is passing through points this A, B, C, let me name it as A, 1 comma 0, okay? This B which is basically 0 comma 0, okay? And this is C which is minus 2 comma 4. So this parabola is basically passing through these points and this is our axis of the parabola, okay? So all these three points should satisfy the equation of the parabola. So let's put that values in this equation and from there we will try to find this value of A, B and C, okay? So our first point is A whose coordinates are 1 comma 0. So it will be basically y is 0, so 0 is equal to this A plus B and plus C. So this will be our first equation. Now let's put this point B. So it will be 0 equal to C. From here we are getting C is equal to 0, right? And C that is minus 2 comma 4. So it will be basically 4 is equal to x square means what? 4A. 4A and minus 2 into B that will be minus 2B and plus C. So this will be basically our third equation, okay? So from here we already got the value of C as 0. So let me write it as, let me rewrite the equation first as A plus B equal to 0. And this will be our 4A, okay? 4A minus 2B, C is already 0, so I'm not writing that C. 4A minus 2B is equal to 4, okay? Then multiply this by 2 and add it. Then it will be this 2A, 2A plus 2B is equal to 0. Now add these two equations, okay? So after adding these two equations what we will get? These two will be cancelled out. So our 6A will be equal to 4. From here we got A is equal to 2 by 3, okay? Once this value of A is got, this A plus B is equal to 0. So B will be our minus of 2 by 3 and C is known to us already. C is equal to 0. Now put the value of A, B, C in the equation, okay? So what we get? The equation of TheraBola is 2 by 3, x square, okay? Minus 2 by 3, x plus 0. So let's take 3 LCM. It will be 2x square minus 2x, okay? So finally what we got? This 2x square minus 2x is equal to 3y. This will be our final equation of TheraBola. 2x square minus 2x equal to 3y, okay? So this option B is correct. So let's move to the next question, this question number. This was question number 12, right? So we have to take now this question number, question number 13. Okay, now it's a subjective questions, okay? Find the equation of the TheraBola whose focus is 5 comma 3 and directrix is the line, this. So we should draw one rough sketch for this. So let me draw this. So this is our directrix, okay? And this will be our access to the TheraBola. Now what we need to find? We know the focus, okay? So this will be our focus whose coordinates are 5 comma 3. And this is our directrix whose equation is 3x minus 4y plus 1 is equal to 0, okay? And now what we need to find? We need to find the equation of TheraBola. So how can we approach this? Like, we can do one thing. We can assume any point on this TheraBola, okay? Let me write its coordinate as h comma k, okay? So this P s will be equal to P m. What I will do? I will draw this thing. So this P s is equal to P m. This is the definition of TheraBola itself. So the path traced by any point whose distance from any fixed point that is focus is equal to distance from any fixed line that is directrix. So P s is equal to P m or we can say this P s square is equal to P m square, okay? So how can we get this length P s? It will be h minus 5, okay? h minus 5 whole square plus k minus 3 whole square. And that will be equal to P m. What will be P m basically? It will be this 3h minus 4k, 3h minus 4k plus 1, whole divided upon this mod thing. This will be under mod under root of 3 square plus 4 square. And this complete thing we have to take. So from here, if you see, from here if we see what we have to do, we have to expand this, right? Okay, then let's open this brackets. So this will be h square plus 25 minus 10h plus k is square plus 9 minus 6k. This whole thing. This will be 5, 5k square purchase. So let's multiply it, cross multiply it. And this thing will be a plus b plus c whole square. So we open it a square plus b square then plus c square plus 2 a b. So it will be 236 into 4 that is minus 24 hk, right? Then plus 2bc, 2 into 4 that is minus 8k. And plus 2ca that will be 6h, right? 2 into 1 into 3h that will be 6h. So this is what we got after opening the brackets and opening this square thing. So let's simplify it. What will be this 25h square minus 9h square, right? So 25h square minus 9h square will be 16h square. Okay, then this k square thing we add it up means we write this k square term. So 25k square and plus 16 that will be 9k square, right? Plus 9k square. Here we get minus 250h, right? Minus 250h, you know, it will be 10h here. So minus 250 and what is here? 6, so minus 256, minus 256h. So this h term is also covered. Now take the k term. It will be 25 into 6 means minus 150. And minus 150, this will be plus. So this will be 142, right? 142k. So these k terms are also covered. Now we are left with this 25 plus 9. That is 25 plus 9 means 34 into 25. So 34 into 25. So 25h is 100, 10. So this will be 25, 75 and 10, 76. So 760. So this will be our 760 here. And what is constant term here? 24k we have not covered. Okay, so that also we have to move it. Let me first move it constant term. So this will be 760, right? 2400. Then 10 will be in carry. 75. Then 85 will come. 75 and 10, 85 will come here. So this will be basically 850. 850 minus 1. And this 24hk will come on this side. It will become 24hk. I think it's clear. 16h square, yeah. So now what we will do? We will replace this h and ky, x and y. Okay. So it will become 16x square plus 9y square. And take this thing 24xy, 24xy. Then what is there? Minus 256x plus 142y. And this 850 minus 1. That is 849 is equal to 0. So this will be our answer. This will be the required equation of parabola. Okay. So hope this is clear to all. And now let's check the next question. Question number 14. It is saying find the equation of parabola whose focus is given. Whose coordinates of focus is this and vertex coordinate is this. So we are having the coordinates of this focus and vertex. And we need to find the equation of parabola. Okay. This will be our directions. This will be our direction somewhere. And this will be our axis. Okay. So what is the focus? So focus of this parabola is at minus 6, minus 6. And vertex is it. This is our vertex. So the coordinates of vertex is minus 2, minus 2, 2. And we need to find the equation of parabola. Okay. So we can do one thing. This is our direct tricks. This is our direct tricks of the parabola. And this is our axis. The axis of parabola. So if I take point, if I take point as a, the point of intersection as a. So what will be the coordinates of a? Suppose I am saying the coordinates of a is what you say this a comma b. Okay. So basically this vertex will be the midpoint of this is. So we know the vertex coordinates so we can find out the coordinates of this a. Okay. So it will be basically a minus 6 upon 2 will be the x coordinate of vertex that is given as minus 2. So from here we get a as minus 4, plus 6 minus 4 means a will be equal to 2. Okay. And for b, this will be basically b minus 6 upon 2 will be equal to y coordinate of vertex that is 2. So from here we get b is equal to 6 and plus 4. So b will be equal to 10. So basically the coordinates of a is coordinates of a is what 2 comma 10. We got the coordinates of a as 2 comma 10. Okay. Now what we can do if we anyhow we can get the equation of direct tricks. Right. So we are having the coordinates of this focus. So if anyhow we can arrange the equation of direct tricks we can take any point on this parabola p. Right. And we can write the equation of parabola like what we did in the last question. So our target is to find this equation of direct tricks. So how can we find this equation of direct tricks? Like our target is to find this equation of direct tricks. So if you see what will be the slope of this axis, slope of axis, slope of axis of parabola it will be minus 6 minus 2. Okay. Upon minus 6 and plus 2. This will be equal to minus 8 upon minus 4 that is 2. Okay. So actually slope of axis comes out to be 2. So our slope of direct tricks will be since both these lines are perpendicular it is making an angle of 90 degree. So our slope of direct tricks will be minus of 1 by 2. Right. Slope of direct tricks will be minus of 1 by 2. And it is passing through a comma b. Okay. So we can easily write the equation of direct tricks as y minus y1 is equal to m x minus x1. So x minus 2. Okay. So it will be basically 2 y minus 20 is equal to this minus x plus 2. So 2 y plus x. Okay. 2 y plus x minus 20 minus 2 that will be minus 22 is equal to 0. Or we can simply write it as x plus 2 y minus 22 is equal to 0. This is the equation of what? Equation of direct tricks. Okay. So we got this equation of direct tricks as x plus 2 y. Minus 22 is equal to 0. Now what we can do? We can take any point on this parabola. Okay. Ph comma k. And we can equate this distances. This p is and p. Okay. From here we can easily get the equation of our parabola. So suppose I am taking this as what you say, p s is equal to p m, right? Now what will be our p s? P s will be h plus 6. H plus 6 whole square plus k plus 6 whole square under root. So better we take squares of this h plus 6 squared plus k plus 6 squared. This is equal to this thing 1 into what will be the perpendicular distance this p m. It will be h plus h plus 2 k. H plus 2 k minus 22 whole thing more upon under root of 1 square, right? Under root of 1 square plus 2 square. Okay. This whole thing. Whole squared. So from here we get what? This will be our h square plus 36 plus 12 h plus k squared plus 36 plus 12 k. Okay. Whole thing multiplied by what is this under root of 5. After aspiring it will become 5. So after cross multiplying it will be whole multiplied by this. And this will be equal to our h plus 2 k, h plus 2 k minus 22 whole square. So on solving this, we can easily find the equation of parabola. So this was the concept in this question. So I am leaving this further expansion of this equation up to you. So please expand it and finally write the equation of the parabola. So I think this is clear, the concept is clear, like what we have done. We were having this vertex and coordinates of vertex and what you say focus, yeah. So we got the equation of directrix by finding the coordinates of A and finding the coordinate slope of this directrix. So after that, we just equated this P s is equal to P m, which is the basic condition for parabola. And after that we got this equation. Okay. So please expand it, please expand it and write it in the simplest form. That will be our final equation of parabola. So let's see this question. Find the vertex, focus, axis, directrix and lattice rectum of the parabola, this. So one parabola is given here. This equation is one parabola is given here, 4 y square plus 12 x minus 20 y plus 67 is equal to 7, right. So this is our given equation of parabola. We have to find the different parameters regarding this parabola like vertex, focus, axis, directrix and all. So let's try to make it a perfect square in y and then we will compare with our standard form. So this will be basically 2 y, 2 y minus 20 y is there, no? So 2 y minus 5 whole square. So this will become 4 y square, then minus of 2 into 2, 4, 5, 20 y it is there, plus 25 we are adding it. So we have to subtract this minus 25. So this is remaining here, plus 12 x and plus 67 is equal to this way, right. So what we got 2 y minus 5 whole square is equal to take this thing to the right hand side, it will be minus 12 x and plus 67 minus 25. It will be 42, 42 means 52, 62, 67, yeah 42. So after coming in this side, this is plus, no? So it will be minus of 42. Further we can write it as 2 y minus 5 whole square is equal to 12. So what we can we take common, we can take 6 common, no? 6, 2 and so yeah. So let's take minus 6 common. So it will be 2 x plus 7, right, minus 12 x and minus of 42. Okay. But here we are having, we are not having the coefficients of this y and x as 1. So for that what we have to do, I am taking 2 common from this bracket. So after coming out of the bracket, it will be in the square form. So it will be basically 4 into this y minus 5 by 2 square is equal to this thing will be minus 6 and take 2 common from here. So it will be x plus 7 by 2, right. So 6 into 12, so this thing will be minus 3. So finally we got our equation of parabola as y minus 5 by 2 whole square is equal to minus 3 x plus 7 by 2. So this is clear to us up to this point. Now, what we will do, we will compare it with our standard parabola, standard equation of parabola that is y square is equal to minus 4 a x. Okay. So what will be the vertex of this parabola. So y minus 5 by 2 is equal to 0. From here we get y equal to 5 by 2 and x if you see x. So x plus 7 by 2 is equal to 0. From here we get x is equal to minus 7 by 2. So this will be our coordinates of vertex. Vertex coordinates, right. So we got the coordinates of vertex. Can we draw a rough sketch for this? y square is equal to minus 4 a x, no. So let's draw one rough sketch. One rough sketch, it will be y square is equal to minus 4 a x means it will be leftward of me, right. There will be some direct text for this. This will be the axis. Okay. This is in which form y square is equal to minus 4 a x. Okay. So yeah, it will be leftward opening only. So our vertex, this is our focus, right. This is our vertex. This is our axis. And this is our directorics. So we got the vertex, we got the vertex as 5 comma 2 and. Sorry, x coordinate is x coordinate is your minus 7 by 2. So the coordinates of vertex is minus 7 by 2 and y coordinate is 5 by 2. Okay. And what next we got the vertex? What will be the focus? What will be focus like what will be the coordinates of s? So if you see this parabola will be axis will be parallel to this axis of this parabola will be parallel to x axis, right. So y coordinate of focus, y coordinate of focus will be same as the y coordinate of vertex. Okay. So I'm first writing the y coordinate and what will be this x coordinate. If you see what will be the value of a what will be the value of a from here from here we can get the value of it. So this thing, this 4 a 4 a is equal to this 3. So a we got the value of a as 3 by 4. Okay. So basically what you say the x coordinate of focus will be this minus 7 by 2 the x coordinate of vertices and minus minus of this a that means minus of 3 by 4. So it will be nothing but this 4 2 times means minus 14 and minus 3 that will be minus 17 by 4 comma 5 by 2. So this will be our this will be this was our vertex and this will be the coordinates of our focus. Now what will be the axis axis is nothing but axis will be parallel to x axis, right. So it will be equal to y equal to some constant that is nothing but y is equal to this coordinate axis. So y coordinate of this either you can take parabolic vertex or focus. So y is equal to 5 by 2. Let me write here also this coordinates this minus 7 by 2 and 5 by 2 and the focus will be minus 17 by 4 comma 5 by. Okay. So axis will be basically line parallel to x axis. So axis will be y is equal to 5 by 2. So we got this vertex this focus this axis. Now, what is the directrix? Okay. So for directrix for directrix for this type of parabola y square is equal to minus 4 a x right this directrix will be parallel to this will be parallel to y axis right. In this in this particular case the directrix will be parallel to y axis or you can say perpendicular to x axis since x axis line parallel to x axis in the axis. So directrix will be parallel to y axis and its value will be this if you see this vertex kajo y coordinate at the y coordinate of vertex and plus this a that means y is equal to this 5 by sorry not y equal to x equal to if I have the same like if I have said y equal to constant I was wrong actually it will be x is equal to some post not y is equal to some post it will be x equal to some constant. So I'm taking my words back if I have mistakenly spoken like y equal to constant. So directrix will be parallel to y axis and it will be the equation of directrix will be x equal to this thing this minus 7 by 2. And plus a what is a plus 3 by 4. So this will be basically 4 minus 14 minus 14 plus 3 that is minus 11 by 4. So this will be our directrix x is equal to minus 11 by 4 will be our directrix. So we found this directrix also now what lattice rectum okay so lattice rectum is nothing but lattice rectum the length of lattice rectum is 4 times a 4 times a so that will be 4 times what is a is 3 by 4. So it will be equal to 3 units. Okay. So hope this this is clear to all say like one equation of parabola is given and question is asking about the different parameters of the parabola like vertex focus axis. So we have completed all these things. So we got this vertex focus axis directrix and lattice rectum. So this was our question number 15 let's say the next one and the name of the clinic represented by this equation. Okay. So one equation is given here this x by a okay under root then plus under root of y by p is equal to 1. Okay. So first let's aspire it. So after aspiring what we get x by a plus y by b plus two under root two times under root of x y by a b right. This whole thing will be under root and is equal to one right now what I will do I will move this to right hand side and again I will spare it. So this will be x by a plus y by b. Okay. And this thing will be one minus two times under root of x y upon a b. Okay. Okay. Now we can further spare it. So what we get we will have x square by a square plus y square upon b square. Okay. Plus two times x y upon a b is equal to one plus four times x y upon a b right a square plus b square minus two a b. Minus two one into a since that will be basically four times four times under root of x y upon a b right. This will be what we get or we can we could have done one thing like to avoid this thing what we could have done. We could have taken this one to the left hand side and we can we could have taken this to root x y by a b along to the means that term alone is in the right hand side that would be better I think so let's try in that way. Okay. So I will take this one to left hand side and I will take this is the root thing to our right hand side. So our calculation will be easy. So x y a plus y by b minus one. Okay. This thing and this thing I'm taking to right hand side. So this will be two under root x y upon a b. Now a square both both hands sides. So what we will get it will give me x square upon a square a square plus b square. Okay. Plus c square plus two a b that is two x y upon a b plus two b c that is minus two y upon b plus two c a that is minus two x upon a. Okay. This is what we got from left hand side and it will be basically four times x y upon a b. Okay. Now, simplify it take LCM. So this will be a square b square. So we got x square b square plus this a square y square. Okay. And what is this will be a square b square. Two a b x y. Okay. And this will be b no. So a square b two a square b y and minus two a b square x right. This will be equal to four x y upon a b. So we can cancel this a b with one a b. Okay. And now we can cross multiply it. So it will be basically b square x square. Okay. Then two a b x y. Then a square y square. Okay. What else minus two a b square x minus two a square b y. And this constant down this what a square plus b square. Okay. Plus a square b square is equal to four x y four x y into a b. Okay. Or four a b into x y you can say. So I'm taking this to left hand side and. Okay. And I'm writing it directly here. So this will be basically minus of two a b. This thing will become minus of two a b into x y and this will. After taking into taking it to left hand side. So it will be minus of two a b x y is equal to zero. Okay. So this is what we got after doing all this. Now, let me remind you like what is the general equation. Like question is asking to find which conic this equation represents fine. So what is the general equation of second degree for me. General equation. Oh, second degree for me. Oh, everyone is aware with this general equation of clinic. It is nothing but a x square at a x square. Plus two h x y. Okay. Plus b y square. Plus a x square plus two h x y plus b y square plus two g x. Plus two f y plus say is equal to say right. For here for this general equation of second degree. We are having different conditions for which this equation represent different onyx. Like if delta is not equal to zero. If delta is not equal to zero. Now what is delta delta is nothing but this value a b c plus two f g h minus a f square. Minus a f square minus b g square minus c h square right. So this is our delta if delta is not equal to zero. And our h square this thing is h square is equal to a b if delta is not equal to zero. And our h square is equal to a b if these two conditions are satisfied by any of the second degree clinic. Then it means it will represent a parabola. Okay. These two conditions. These two conditions if satisfied by equation. If satisfied by this general degree equation e. Okay. If satisfied by e it will represent it will represent a parabola. It will represent a parabola. Okay. So from here if you say from here. So this is the first condition this is the second condition. So what is h here. If you see here the value of h will be value of h will be a b. Right. Means two h if you consider if you compare this two h is equals to two a b right. So our h is coming out to be a b. And what is this capital A and capital B capital A is nothing but the coefficient of x square. Okay. That is B square in this case. And what is B capital B capital B is nothing but the coefficient of y square which is a square in this case. So here if you can see h square is equal to h square is equal to a b. So this condition two is getting satisfied. Right. And this if you calculate this value of delta that is a b c plus two f g h minus a f square minus b g square. Okay. I've written here in the short form. Right. No, but here I've written it in capital form. So let me write it in this way. Otherwise it may create a misunderstanding to all of you. So this will be a b c. Okay. This delta will be a b c. Okay. Plus two f g h. Okay. Two f g h plus b y square. No. A b c plus two f g h minus a f square. So this will be minus of a f square minus of b g square and minus of c h square. Okay. So two f g h. Yeah. Okay. Let me write it as capital. So this delta a b c plus two f g h minus a f square minus b g square minus a h square you put you compare this equation with this and you will get the value of delta to be non-negative. Hence, this equation basically represent delta is equal to delta will be coming out to be non-zero and h square will be equal h square is equal to a b which have already seen new. So this equation basically represents a variable. Okay. You please verify by putting the different values of this a b c and you will be having the value of delta will be coming out to be non-negative. Sorry. Non-zero. Okay. So this is all for this question number 16. Now let's take this question number 17 similar question. Determine the name of the curve described parametrically by the equation this. Right. So the x and y coordinates are given in parametric form. So our x is given as t square plus t plus one. Okay. And our y is given as t square minus t plus one. Okay. So this x and y are given in parametric forms and now we have to represent like the curve. This parametric form will generate what will be the name of the curve whose parametric form is given by this. So let's try it like let's add both these equations. So what we will have, we will have this x plus y is equal to two t square plus two. Okay. So two t square plus two. And two common. So it will be two t square plus two. So this will be one thing. And now one more thing you can do. You can subtract it also. So it will be x minus y is equal to this thing will be cancelled out. This one will also cancel out. This will be two t. Okay. So this three questions what we got. Now what I will do from here. We got the value of T as x minus y upon two. Okay. We will substitute this value of T in equation one and we will solve further. So what I will do, I will substitute this value here. So we will get x plus y is equal to two into t square. So this will be basically x minus y upon two whole squared plus one. Okay. This is what we get. So x plus y will be equal to, this will be four x square plus y square minus two x y. Right. Two x y plus four. And this is x plus y anyhow. So this will be two times. So this after cross multiplication, it will become two x plus two y is equal to this square plus y square minus two x y and what plus four. Right. This is what we will get. So finally it will become x square plus I'm doing here x square plus y square minus two x y. Okay. Minus two x minus two y. Okay. Plus four is equal to zero. So similarly, what we have done in the last question, we will compare with the general second degree equation and we will find this value of delta delta and the delta should not be equal to zero. Like we will find first value of delta and we will see this h square is coming out to be AB or if these two conditions satisfied means this equation will represent a parabola. So let's check this. Let's check it out in this question. Like last question, I have left it to you. So let's see here. So in this equation, if you see what will be our A. Okay. I'm writing in the short form. Otherwise it's creating confusion. So this will be h square is equal to AB. These two conditions for parabola we will check one after the other. So first tell me what will be A. So A will be equal to one. Okay. What will be B? B will be equal to one. Two h is equals to two h is equals to minus two, right? Two h x y. So from here we get h is equals to minus one. So A is known to us. B is known to us. Two g x. So two g is equals to minus two. So g will be equal to minus one. Two f is equals to minus two. So f is also minus one. And c is equal to four. Like in this question, I'm doing it so that you can see how we get this value of delta and this h square is equal to AB or not. How we used to compare. So if you say delta is nothing but ABC. Okay. Plus two f gh minus a f square minus b g square minus c h square. Right. So this is ABC means what? A into B into C that will be four plus two f is minus one. G is minus one h is also minus one and minus a a is one f square means it will be one minus B B is one and G is where it will be one minus C C is four. So this will be minus four one h is minus one. So this is coming out to be four. And two, this will be completely minus two then minus six. So from here you see four minus eight minus eight plus four means this is minus one. So delta is coming out to be non-zero. Okay. So this condition is satisfied. This condition is fulfilled. Now what is h square? H square is equal to minus one couple square that is equal to one and AB. AB is one into one. That is also one. So this H square is equal to AB. So this condition is also getting satisfied. So since these two conditions are getting satisfied, we can say the set of the equation represents there are other conditions also for this general second degree equation upon it. Okay. But since we are dealing with parabola, most probably the question will be asking for parabola only. So I'm checking for parabola, but if this, suppose this delta will be coming out to be zero, right, then it will, it will represent a pair of straight lines. Similarly, there are other conditions also for other colleagues like for a hyperbola and a list that we will take care in the coming sessions. Okay. So this equation is representing one parabola that is for sure. So this is done. Question number 17. Okay. What is this question asking prove that the equation of parabola, who's a vertex and focus on the x axis at a distance of a and a dash from the origin respectively is this. Okay. Let me draw one for this case for this. This is our. This is our. This will be our. Okay, let's see. So it is saying that equation of parabola was vertex and focus on the x axis. Okay. Means this is our x axis basically. This is our x axis. This is our y axis. Okay. And suppose this is our focus is this is our vertex B. Okay. So it is saying that for vertex and focus are at a distance a and a dash from the origin. Okay. So suppose this is our origin. So this distance is given as a right distance from O to this V is given as a and this distance is given as a dash. This is what the question is saying was vertex and focus around the x axis. Yeah, it is there at a distance a and a dash from the origin. Then prove that this will be the equation of parabola. Okay. So basically what will be our coordinates of this is coordinates of s will be basically a dash comma zero. Okay. And this coordination coordinate of this V will be a comma zero coordinate of V will be a comma zero. Right. And what will be this distance? This V is this V is will be nothing but a dash minus a a dash minus a. Okay. So any parabola whose axis is parallel whose axis is parallel to x axis. Right. So how do we write that equation? We used to write it as y square is equal to 4 a x. Okay. Or let me write it in this way. Y as capital. So we usually write it as in this format. This is the standard parabola whose axis is parallel to x axis. Right. And whose axis is parallel to x axis and whose vertex is at origin or zero comma zero. Now, in this case, in this case, if you see what is this basically this 4 a represents 4 a or what is this a represents a represents nothing but this distance between what is a distance between distance between vertex and focus well vertex and focus. So in this particular case, what is given in the question? What is a is nothing but a dash minus a a dash minus a right. And how much is this what we say? This is what is the vertex here? Vertex is a no vertex for this parabola is a coordinate of vertex is a comma zero. So this x will be replaced by x minus a right. No, it's coordinate will be replaced by this x. This thing is x minus a is equal to zero means x is equal to a that is the x coordinate of vertex that will be x coordinate of vertex which is known to us a comma zero. Okay, so in this equation what we have to do in this equation we have to replace this y square will be as it is for will be as it is a will be replaced by this a dash minus a and this x will be replaced by x minus a. This is what the question is saying so definitely this will be the equation of the parabola for the given conditions. Right. So this was our question number 18. I think one or two questions. 1920. Yeah, two questions are there. Okay, we will close it. We will close it. So what is saying in question number 19. Parabola, whose axis is parallel to x axis, right, and which passes through these points also find its lattice vector. Okay, so better to throw out of a sketch. Find the equation of parabola whose axis is parallel to x axis. No, okay, so. Parabola will look like something. This, this is the direct tricks. And this is the axis of the parabola. Okay, what is information is given its axis is parallel to x axis. Okay, and it is passing through the points this this and this. Okay, so parabola is passing through these points. Okay, then let's consider this point as a was coordinates are zero comma four. Okay, and let's take this coordinate as this point as B was coordinates are one comma nine. Let's take this as C was coordinates are minus two comma six. So this is our axis, which is parallel to x axis. This is what is given in the question. Okay, and it is passing through these three points. So we have to write the equation of parabola. Okay, so how can we do that? Like any parabola, right? Any parabola whose axis is parallel to x axis. This is what is given in the question. We can write it in this form. This x is equal to this a y square plus b y plus equal to zero. No, x is equal to a y square plus b y plus. Right, this parabola will represent represent parabola. Okay, this equation will represent parabola whose axis is whose axis is parallel to x axis. Right now, since these three points through these three points, parabola is passing. It means it will satisfy the equation of parabola. So this point a that will be zero comma four. If we put here what we will get, we will get four is equal to not for x is zero. So x is zero means zero is equal to this four squared that will be 16a. Okay, 16a plus four b and plus c. So this is our first equation. Now let's put this point b. So it will be x is equal to one nine squared means 81 a plus nine b plus c equals to zero. This will be our second equation. This point c minus two comma six. Right. So minus two will be equal to six squared means 36 36 a plus six b and plus c is equal to zero. So we are having three equations, three angles, three equations and three unknowns. So we can definitely find it out. Okay, after finding the value of ABC, you put in this equation, that will be our answer. Right. That will be our answer. So I'm not going to solve this three equations we have made in terms of AB and C right and three unknowns are there. So it we can be easily solved it out. So please do it and put the value of ABC in the given equation that will be our answer that will be our equation of the parable. So, okay, this is our last question question number 20 first exercise. So this equation is a x square plus four x y plus y square plus a x plus three y plus two is equal to zero. This equation represents a parable. Okay, the question itself says that this equation is representing a parabola, then we have to find the value of a. Okay, so we know here, this two H is equals to two H is equals to four. So H is equals to two. Okay, now what is the value of a here, a is the coefficient of x square. A is equal to a here, we will take it and what is the value of B, B is the coefficient of y square that is one. Right. So for this equation to represent a parabola, this H square must be equal to AB. Right. Now what is H square that is four and a we have to find the value of a B is one. So from here, we got the value of a is four. So A must be equal to four for representing this equation to be a parable. So I think we are done with this exercise. I think the video is also long for this particular exercise. Anyhow, we can't kill. So this is all for today. We will come back soon with our next exercise on parabola that is exercise two. So till then, ta ta goodbye and you all please take care of yourself.