 Next is the concept of radical axis, next is the concept of radical axis, okay. So what is radical axis? So radical axis of two circle is basically, this is defined as radical axis of two circles is the locus of a point, is the locus of a point which moves in such a way, which moves in such a way, which moves such that we can write, which moves such that, the lens of the tangent, the lens of the tangents drawn from it, drawn from it to the two circles, to the two circles are equal, are equal. So let's say these are the two circles, okay. Now the locus of such points from where you can draw two tangents of equal length. So let's say there's a point here, P, from where the tangents drawn to these two circles are equal in length. Then all such points, all such points from where you can draw equal tangents to these two circles would be called as the radical axis, would be called as the radical axis. Now let us derive the equation of this, again it can come as a direct locus based question to you, they may not use the word radical axis per se. But let's say the equation of this circle is x square plus y square plus 2g1x plus 2f1y plus c1 equal to 0 and this circle is x square plus y square plus 2g2x plus 2f2y plus c2 equal to 0. Then I have to find the locus of P point, which is h comma k. Can you find, can you guys give me the equation of the radical axis now? Treat this as a locus question. By the way, if this equation is s dash equal to 0 and this is s double dash equal to 0. What is the length of the tangent? Length of the tangent is s1 under root s1 dash and this will be under root s1 double dash. Absolutely correct, tapas, very good. So when you use these two, basically what I am trying to say is that under root of s1 dash is equal to under root of s2 dash, double dash, correct. So if you simplify it, it simply becomes s1 dash is equal to s1 double dash, correct. Now this s1 is because you are putting the point h comma k. If you generalize it, it just becomes s dash equal to s double dash, correct. Which is as good as saying s dash s double dash equal to 0. That is what tapas wrote, which is actually nothing but 2 g1 minus g2x plus 2 f1 minus f2y plus c1 minus c2 equal to 0, okay. Basically you would realize that it resembles, it resembles the equation of common chord and is the reason why it happens, why the resemblance is there, I will tell you the reason. See, if these two circles intersect each other, right, you would realize that the common chord itself, the common chord itself would actually be your radical axis because if you extend this outside the circle, if you extend this common chord outside the circle, okay. So outside it would be known by the name of radical axis and within it it would be known by the name of common chord. So the same line is playing two roles. It is playing the role of common chord and it is also playing the role of radical axis. When does it become common chord, when it is within the circle, when does it become radical axis, when it is outside the circle, okay. So any point, from any point on this line when you draw tangents, they would be equal in length, they would be equal in length. So when you separate them out, there is no existence of common chord but yes, the existence of chord of radical axis will still be there, the existence of radical axis will still be there. Now guys, I will not start with the properties of radical axis because we don't have much time to discuss that. We at least need half an hour to do that. But at least we can take one question based on radical axis, okay. Let's take this question. Let's say we have three circles, one is this, another is this and another is this. Let's say this is S dash, this is S double dash, this is S triple dash, okay. Find the point, find the point, find the point from where the tangent drawn to the above three circles will be equal in length, will be equal in length. Question is clear. So you have three circles, find the point from where tangents, tangents drawn to these three circles will be equal in length, tapas says 3, 2. What about the others? Atme says 3 by 2, half, okay guys. So basically, do you all recall that the point from where you can draw three tangents would lie in the, the point will lie on, point will lie on the radical axis, radical axis of S dash and S double dash, right. And also the point will lie on the radical axis of S dash and S triple dash, right. Any two pairs you have to, you can take up, okay. So what's the equation of the radical axis? Equation of radical axis of S dash and S double dash. So subtract these two equations, so you'll get 4x minus 4y minus 4 equal to 0, that means x minus y minus 1 equal to 0, that's the first equation. Similarly, equation of the radical axis of S dash and S triple dash would be minus 2x plus 10y minus 14 equal to 0, that means you can say minus x plus 5y is minus 7 equal to 0, correct. Yes or no? Add them. So you get 4y minus 8 equal to 0, so y is equal to 2, so y is equal to 2. So if y is equal to 2, x will be equal to y minus y plus 1, which is equal to 2 plus 1, which is 3. So the point is going to be 3 comma 2. Absolutely correct, Tapas, Sayuja, Lalitha, Abnesh, better luck next time. So guys, thank you so much for coming online over and out from Centrum Academy. So next session would be a little bit more about the radical axis property and then we'll be starting with the concept of parabola, okay.