 Next is the concept of intersection, intersection of a line and a circle, intersection of a line and a circle. So let me draw a circle quickly over here. So let's say this is a circle and let's say we have a line, then this line and this circle will interact in three ways. One is this, another is like this and another is like this, okay? So there can be three situations. The line may cut the circle as in A or touch the circle as in B or not even touch the circle as in C, okay? So now let the equation of, I'll take a very simple case, let the line be y is equal to Mx plus C and let the circle be the standard case of a circle, that means having its center at origin and radius as A, okay? Now in order to know the condition for A, B and C what I will do, I will simultaneously solve the equation of the line with the circle. So I will simultaneously solve this. So I would replace y with Mx plus C in the second equation which will end up giving me this quadratic equation and if this quadratic equation has, so let's say I call this one, so if one has real roots, okay? If one has real roots then it will lead to the case A, okay? If one has, sorry, real and distinct roots I should say, real and distinct roots it will lead to condition A. If one has real and equal roots it would lead to condition B that is the line would touch the circle and if one has imaginary roots it will lead to your condition C, correct? The real root means B square minus 4ac should be greater than 0 that will give you 4M square C square minus 4, 1 plus M square C square minus A square would be greater than 0, right? So if you simplify this you will end up getting A square, let me drop the factor of 4 from everywhere, so if you drop the factor of 4 you will get M square C square minus this greater than 0, expand it, okay? So these two terms will get cancelled, so you will get C square lesser than A square plus A square M square that is your C square will be less than A square 1 plus M square for you to have real and distinct roots. So this is your case A, so this becomes your case A, correct? Similarly without wasting much time I can say for case B your C square will become equal to A square 1 plus M square, okay? This condition is of importance to us because we call this as the condition of tangency we call this as a condition of tangency for a line y equal to Mx plus C to be tangent to the standard case of a circle x square plus y square is equal to A square and finally you can say situation number C where C square will be greater than A square 1 plus M square is that clear, okay? So now we will discuss more about the second condition over here that is the condition of tangency, so again let me first make a circle over here, so this is the standard case of a circle, now this is called the point of contact, P is called the point of contact. Now if I mention that this line has a slope of M, this line has a slope of M and it happens to be tangent to this circle, okay? What could be the possible equations of this line? So you can say that there can be two situations for a line having a slope of M to be tangent to this given circle, one can be a line as shown to you and another can be a line like this, right? Both these lines will have the same slope of M, correct? So how do we get these two equations? How do we get the equation of line root plus minus A under root 1 plus M square, okay? That means the equation of L1 and L2 can be written as y equal to Mx plus minus A under root 1 plus M square, okay? Guys, this is called the slope form of the equation of a tangent, slope form of equation of a tangent to this circle, now can you guys help me to get the point of contact? How do I find P and Q that is the point of contact of this line with the circle, okay? So we are talking about the condition of tangency and I just now explained that if you have been given a slope that is M with the same slope and the given equation of a circle that is x square plus y square equal to a square, you can draw two tangents, one at P and one at Q, both will be as you can see parallel to each other and they would be at the diametrically opposite ends, okay? And the equation of L1 and L2 is given as y equal to Mx plus minus, plus minus signifies that there can be two equations and then I had asked you to give me the coordinates of the points of contact, so P and Q here are the points of contact, okay? So all of you please work out the point of contact, give me the point of contact in terms of A and M, only in terms of A and M, what would be the points of contact P and Q? So please quickly work it out and tell me the result, in fact I will be helping you in one minute time, so please solve it on your own first, so we wanted this point of contact. So again situation is very simple, if you again substitute y as Mx plus C in the equation of the circle, you are going to get something like this, x square 1 plus M square plus 2 Mcx plus C square minus A square equal to 0, okay? And the moment you are going to substitute C, okay? So let's substitute C first of all as A under root 1 plus M square in this. So we will end up getting this and C square minus A square will be like A square M square. So basically this is what we get to see, which is actually a perfect square, this is a perfect square, okay? So from here if we get the value of x, we get it as minus Am under root of 1 plus M square, okay? Now this is the value of x, what will be the value of y? Similarly put this in the equation of a line y equal to Mx plus C, okay? So using the value of x as minus Am under root 1 plus M square and C is A under root of 1 plus M square, taking the LCM, you get minus Am square plus A1 plus M square, which is going to be A by under root of 1 plus M square, okay? So one of the point of contact would be minus Am under root of 1 plus M square comma A by under root of 1 plus M square. Now this is obtained when I assume C to be this. If you assume your C to be minus A, if you assume your C to be minus A under root of 1 plus M square, you will get a point of contact like this, Am under root 1 plus M square comma minus A by under root 1 plus M square, okay? In other words, in other words, the coordinates of P and Q would be minus plus Am under root 1 plus M square comma plus minus plus minus A by under root 1 plus M square, okay? Many books remember it like this. So they put an A square and put an A over here, A square and then over here and they remember it like this, minus plus A square M by C comma plus minus A square by C, okay? That's a memory aid for you so that you don't have to remember the big term. You already know what is your C actually. Is that fine? So this is the equation of the tangent in the slope form. Now we'll talk about different forms of the equations of the tangent. After this, I'll give you a break for five minutes, okay? So first form that we are going to discuss is the point form. The point form of the equation of a tangent. So just now we discussed slope form. In fact, let me just list it out over here. So we discussed slope form a little while ago and slope form, its equation is y equal to Mx plus minus A under root 1 plus M square. Why it is called slope form? Because here M will be given to you, slope will be given to you, okay? Next form is the point form. Let me call it as the second point form. In point form, you would be given the equation of the circle. So let's say this circle is given to you, okay? And you have been given that at the point x1, y1, you are drawing the tangent. Now this was a case when the equation of the circle was x square plus y square is equal to A square. You cannot use it for all the cases, correct? So here also I will start with the same equation x square plus y square is equal to A square. The equation of this tangent as I have already discussed with you can be obtained by simply replacing x square with xx1, y square with yy1, okay? So when you do these replacements, you get the equation of a tangent at that point x1, y1 in a form which is called the point form of the equation of a tangent, okay? And let me tell you this expression, we normally call it as t, okay? Now, even if your circle is a case of a general form of the equation of a circle, so let's say you have a circle which is actually your general form of the equation of a circle x square plus y square plus 2gx plus 2fy plus c equal to 0 and you are drawing a tangent at a point x1, y1, we do the same stuff, we replace or we do the following replacements, we replace our x square with xx1, y square with yy1, x with x plus x1 by 2, y with y plus y1 by 2 and constant remains constant. So the new equation of a tangent drawn would be xx1, yy1, gx plus x1, fy plus y1 plus c equal to 0, this is called the point form equation of a tangent. This process is to be only applied when your curve is a second degree general equation, so for any conic it will work, but it is not going to work for a third degree, let's say equation, okay? So don't start applying it to any other form, it only works, it only works for second degree equations, okay? Moving on to the third form of the equation of a tangent which is called the parametric form, which is called the parametric form. Again, let's say we have a circle, let me take a picture of the circle because we are using it very, very frequently, okay? And you have a tangent drawn to this circle at a point, let's say this is my x square plus y square is equal to a square circle and you are drawing it to some point whose parametric expression is a cos theta comma a sin theta, then you just have to do one thing in the point form of the equation of a tangent, you replace your x1 with a cos theta, replace y1 with a sin theta and you end up getting the equation of a tangent in parametric form as x cos theta plus y sin theta equal to a square, is that clear? Okay, sorry, is equal to a, a square is equal to a. So these are the three equations of the tangents that you need to know.