 Next we are going to talk about parametric equations, parametric equations of standard parabola. I think earlier also I had spoken about the parametric equations of circle and this time I am going to talk about the parametric equation of a parabola. So what is a parametric equation? Parametric equation is basically a equation where instead of directly relating x and y by this relation, you relate it with the help of a parameter. So for y square is equal to 4ax, you can say that the same equation could be written as or it is equivalent to saying x equal to at square y is equal to 2at where t is a parameter, where t is a parameter. That means if you keep changing your parameter t, you keep getting different, different points on the curve y square is equal to 4ax, okay. So in other words, if you try to eliminate your t by using these two equations, you will end up getting the equation of the parabola y square is equal to 4ax, okay. What is the use of a parametric form? The use of a parametric form is in choosing a point on the curve. So if I have asked to choose a point on the curve y square is equal to 4ax, let's say I ask you to choose a point on this curve y square is equal to 4ax, okay. So I would prefer choosing it as at square comma 2at rather than choosing it as x comma y or rather than choosing it as x1 comma y1. So parametric form really saves a lot of time because we use only one variable or one parameter here which is t, okay. In a similar way, if I ask you for y square is equal to minus 4ax, can you suggest me a parametric form? Please type it on your screen. Any parametric form that you can suggest for y square is equal to minus 4ax, take a clue from number one and a response guys. So in this case you would say x equal to minus at square and y is equal to 2at, exactly, correct Shia, okay. Let's talk about x square is equal to 4a y, x square is equal to 4a y. In this case you could say x is equal to 2at and y is equal to at square and similarly for x square is equal to minus 4a y, your x will be minus, sorry, x will be 2at and y will be minus at square, okay. So if you have understood this parametric very well, let me ask you some more questions. Write the parametric form for, write the parametric form for x minus 2 whole square is minus 4y plus 3, great, Arya man, done, please feel free to type in your response in the chat box. So here you'll write your answer as 3 is minus at square, minus a, a over here is 1, so a is 1, so you'll say minus t square and x minus 2 is equal to 2at, 2at means 2t, okay. So your parametric form would be y is equal to minus 3 minus t square and x is equal to 2 plus 2t, x will be equal to 2 plus 2t, yeah, correct, correct. So now we are going to move on to the next concept which is position of a point with respect to a parabola, position of a point with respect to a parabola. So let me take a case of a standard parabola. Now a point can lie at three positions with respect to a parabola, let's say there is a point x1, y1, okay, now this x1, y1 point can either lie at A, can either lie at B or could lie at C, or could lie at C, right. Now how would I come to know whether this point is lying within the arms of the parabola, that means at position A or it is lying on the parabola which is at position B or it is lying outside the parabola which is at position C, okay. So for that what do we do is, let's say the parabola is y square is equal to 4ax, okay. Now if the point is at A then y1 square would be lesser than 4ax1, okay. If the point is at B, y1 square will be exactly equal to 4ax1 and if the point is at C, y1 square would be greater than 4ax1, okay. Now this is something which I want to introduce right now in conic, in fact I would do the same thing in case of circle and other conics as well, normally this y square minus 4ax term we call it as s, s stands for second degree terms, okay. So we name this y square minus 4ax as s, so it's a name which is assigned to it, okay. And when you put a point in place of x and y, we call it as s1. So s1 is y1 square minus 4ax1, okay. So remember these two terminologies that we use in case of all conics. So the same situation A, I can write it as s1 less than 0. Condition B could be written as s1 equal to 0 and condition C could be written as s1 greater than 0, where s1 is also called as power of a point with respect to the parabola, power of a point with respect to the parabola. So it's something which is the name which is given to this s1, s1 is called the power of a point with respect to the parabola. So this power is negative which means the point is inside the parabola, that means within the branches of the parabola. If power of a point is 0 with respect to a parabola means it is on the parabola and if the power of a point is greater than 0, that means it is outside the arms of the parabola. Guys, let me tell you we have already completed the school syllabus for parabola, we are doing over and above the school syllabus, okay. So this is a huge chapter, I will not be going to great depth, I will be just talking about few necessary things, again I will be covering this topic later on in detail. So next concept that we are going to talk about is intersection of a line, intersection of a line y equal to mx plus c with the parabola y square is equal to 4x, intersection of a line y equal to mx plus c with the line with a parabola y square is equal to 4ax, okay. So let's say we have the parabola y square is equal to 4ax, okay. Any line, any line can interact or intersect with this parabola in 3 ways. It can either cut the parabola, right, it can touch the parabola, correct or it may not even touch the parabola. So 3 situation arises, situation number a, situation number b and situation number c. So now let us find out the condition for a to happen, b to happen and c to happen, okay. Now here just like the circle we cannot use the distance from the origin, right. So what concept do you suggest will help us to find out the condition for the line y equal to mx plus c to cut, cut, the word is cut over here, to cut y square is equal to 4ax parabola. Any suggestion guys, please type it in your chat box. So how would I come to know how does this line interacts or intersects with the parabola, okay. So I will help you out in this case, we will make use of quadratic equations for that. How? So we will simultaneously solve, we will simultaneously solve y equal to mx plus c and y square is equal to 4ax, okay. So we will substitute this in place of y. So I get mx plus c whole square is equal to 4ax, correct. If you expand it, this is what we get and form a quadratic from it, form a quadratic from it like this, okay. Now if condition a has to happen, then there must be real and distinct roots of this quadratic equation 1, real and distinct roots should exist for this quadratic equation 1. That means b square b square minus 4ac should be greater than 0, correct. Pull out a factor of 2 out which is 2 square, so it will become 4mc minus 2a the whole square minus 4m square c square greater than 0, okay. Drop a factor of 4, it will become mc minus 2a the whole square minus m square c square greater than 0. Expand it, right, correct RMR. So we will have something like this situation coming up which means amc is less than a square which means, which means m is less than a by, which means c is less than a by m. So this is the condition for the line to cut. So this is the condition for your condition number a to happen, okay. So without wasting much time I can say for b to happen, you should have real and equal roots, real and equal roots, correct. Which means c should be equal to a by m, c should be equal to a by m. This condition is called the condition of tangency. This situation is called the condition of tangency for this parabola y square is equal to 4ax and the line y equal to mx plus c only. Please remember that, okay. So I will write it down. Only for y square is equal to 4ax and the line y equal to mx plus c. If this parabola changes, then the condition will be something else, okay. And for situation number c to happen, you should have non-real roots. You should have non-real roots. That means c should be greater than a by m, c should be greater than a by m.