 Let us take another problem, find the equation of the parabola, find the equation of a parabola Whose focus is at 4 comma minus 3 and vertex and vertex is going to be 4 comma minus 1 So guys mostly the question that you are going to get in school exams would be centred about these kind of questions So they will give you focus, directrix, they will give you the focus and vertex So please try this and feel free to type it in the chat box So always draw a diagram, from the diagram we will get a clear cut indication how is the parabola actually oriented So 4 comma minus 1 will be in fourth quadrant And 4 comma minus 3 also is in fourth quadrant So it is a kind of a parabola which looks like this So it is a parabola which looks like this So from here, can I get the equation of the directrix? So if I draw a line connecting the vertex, let us say this is our directrix So can you get the equation of this directrix? So focus, we know it is 4 comma minus 3 This is 4 comma minus 1 Now if you would realize that this line is actually parallel to the y axis If you see this line is parallel to the y axis So your directrix should be parallel to the x axis Is parallel to the x axis Because we know that the axis and the directrix always make a 90 degree with each other So if all I need is to get this point If I could get this point, my job will be done of finding the equation of the directrix And that is pretty simple because if you call this point to be some point alpha comma beta Then you know that this distance and this distance are going to be the same So alpha plus 4 by 2 is equal to 4 which means alpha is equal to 4 And beta minus 3 by 2 is minus 1 which means beta is equal to 1 So this point is actually 4 comma 1 So the equation of the directrix becomes y equal to 1 And the moment I get the equation of the directrix My life becomes very easy because I have to follow the basic definition of the locus And what is that? Its distance from this point should be same as the distance from this line So these two should be the same However it doesn't appear in the diagram though So let's apply that So h minus 4 whole square y plus 3 whole square under root should be equal to This distance I can directly comment that would be k minus 1 mod Because from here to here it's k From here to here it's mod k And from here to here it's 1 So you can say k minus 1 is the distance between these two This point and the directrix Mod I have taken because in case k is positive or negative It will account for the positivity of the distance Now squaring both the sides On squaring both the sides we get this expression And let's simplify this So it's going to be h square minus 8h plus 16 Plus k square plus 6k plus 9 Is equal to k square minus 2k plus 1 k square and k square gets cancelled So you have h square minus 8h Plus 8k plus 24 Equal to 0 And on generalizing it We get the required equation of the parabola as x square minus 8x plus 8y plus 24 equal to 0 Alright So let's take another question Let's take another question This question is a very very good question Where you would realize actually the deeper meaning of the Equation of a parabola So the question goes like this Find the equation of the parabola Find the equation of the parabola Whose lattice rectum Whose lattice rectum Is four units The tangent at the vertex And the tangent at the vertex Is the line 4x minus 3y plus 7 Equal to 0 Okay So guys the first thing that would arise in your mind What is the length of the lattice rectum That is something which we haven't discussed so far But let's discuss it right now What is the length of a lattice rectum Okay so let's say we have a parabola like this I'm taking a generic case of a parabola Or a standard case of a parabola y square is equal to 4x And to the point a comma 0 I'm drawing a double ordinate I'm drawing a double ordinate Double ordinate is a line which is perpendicular To the axis of the parabola And since this double ordinate is passing through focus It is a focal chord which is a double ordinate So lattice rectum is nothing but a focal chord Which is also a double ordinate Now I want to find the length of this particular line Okay so I want to find pq Now it's very simple If you see this point This point will also have the x coordinate as a Now y coordinate is what I don't know right now And this point q will also have x coordinate as a And what is your y coordinate I don't know right now Okay I will try to figure out these question mark And this double question mark By using the fact that This point p and q Will satisfy the equation y square is equal to 4ax Right So in place of x I'm going to put a now Which gives me y square is equal to 4a square Which means y is plus minus of 2a Plus minus of 2a Indicating that this point Is actually 2a And this point over here is actually minus of 2a Okay So the length pq is just going to be The distance between the point p and the point q Is a minus a square 2a plus 2a square under root Which is just going to be 4a units Okay So remember the length of the lattice rectum Is nothing but 4 times the distance of the vertex From the focus Okay So it is nothing but 4 times the distance Of the vertex From the focus Latticectum as we have already discussed And I will prove this later on It is the shortest focal chord It's the shortest focal chord In a parabola Okay Now guys I want all of you to pay a lot of attention over here Because I'm going to discuss something Which is very very important Right now in this figure See this is your y axis Correct And this is the axis of the parabola This is the axis of the parabola Okay When you look at this equation Y square is equal to 4ax Right You actually should be realizing that this y over here It's actually nothing but It's actually nothing but The distance of any point from the tangent At the vertex Sorry this x is nothing but The distance of any point from the tangent to the vertex And this y is nothing but The distance of any point from the axis of the parabola Correct So what I want you to appreciate over here is that When you say y square is equal to 4ax It actually means Distance Of any point From Axis whole square This equation is same as saying Distance of any point from axis whole square is equal to Length of Length of Lattice rectum Times Distance of Any point from the tangent At the vertex Okay This is a broader definition of a parabola If you look at this y square is equal to 4ax It actually fits into this broader definition So distance of any point from the axis of the parabola Is nothing but Y Correct So if I take any point x comma y Resistance from Resistance from the y axis is Y Right And length of the lattice rectum is nothing but 4a And distance of any point from the vertex is actually x So that makes it y square is equal to 4ax Okay Is that clear guys Now the same concept has been used in this problem as well So as you can see We have been given the length of the lattice rectum which is 4a We have been given the Tangent We have not been given the equation of the axis So please note down the equation of the axis was Equation of the axis was 3x plus 4y minus 4 equal to 0 I think I have missed out on this information So equation of the axis is given to us This is the equation of the axis And this is the equation of the tangent at the vertex Okay So what I will do now I will use this definition To get the equation of the desired parabola So distance of any point from the axis You realize that it would be This mod By Under root of 3 square plus 4 square Correct Square of this That means I am squaring the distance of the point from the axis Okay Equal to length of the lattice rectum which is 4 into I think the given distance is 4 Yeah Length of the lattice rectum is 4 Times distance of any point from the vertex So distance of the same point x comma y from the vertex That would be 4x minus 3y plus 7 Again by under root of 4 square minus 3 square Okay If you simplify it it becomes 3x plus 4y minus 4 The whole square Is equal to 20 times 4x minus 3y plus 7 Okay Again you can further simplify this and get your answer But what I am concerned here is Whether you are able to appreciate This definition of a parabola I think one question in Adi Sharma is based on this Definition of a parabola Very very important Okay This is very very important form of a definition of a parabola