Question

Find the co-ordinates of the point from which tangents drawn to the circle $x^2+y^2–6x–8y+3=0$ such that the mid point of its chord of contact is $(1,1)$.

Answer 1

Let the required point be $P(x_1,y_1)$. The equation of the chord of contact $P$ with respect to the given circle is

$x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c=0$

$\Rightarrow 2g=-6\Rightarrow g=-3$ and $2f=-8\Rightarrow f=-4$

The equation of the chord with mid-point $(x_1,y_1)=(1,1)$ is

$x+y-3(x+1)-2(y+1)+3=0$

$\Rightarrow x+y-3x-3-2y-2+3=0$

$\Rightarrow -2x-y-2=0$

$\Rightarrow 2x+y=3$

Equating the ratios of the coefficients of $x,y$ and the constant terms and solving for $x,y$ we get $x_1=-1,y_1=0$