Question

Find the co-ordinates of the point of intersection of tangents at the points where the $2x+y+12=0$ meets the circle $x^2+y^2-4x+3y-1=0$

Answer 1

$S:x^2+y^2-4x+3y-1=0$.

Let the point of intersection of tangents by $p(h,k)$

Line AB $2x+y+12=0$ is chord of contact

Equation of chord of contact is given by:

$x\left(h\right)+y\left(k\right)-4\left(\frac{x+h}{2}\right)+3\left(\frac{y+k}{2}\right)-1=0$

$\Rightarrow xh+yk-2x-2h+\frac{3y}{2}+\frac{3k}{2}-1=0$

$\Rightarrow x(h-2)+y(k+\frac{3}{2})-(2h-\frac{3k}{2}+1)=0$

Comparing this with given equation $2x+y+12=0$

$\frac{h-2}{2}=\frac{2k+3}{2}=-\left(\frac{4h-3k+2}{24}\right)$

Solving, we get

$h=1,k=-2$

$(1,-2)$.