Question

A circle $S$ touches the line $x+y=2$ at $\left(1 , 1\right)$ and cuts the circle $x^{2}+y^{2}+4x+5y-6=0$ at $P$ and $Q$ respectively. The $PQ$ always passes through the point

• A. $\left(2 , 3\right)$
• B. $\left(6 , - 4\right)$
• C. $\left(- 6 , - 4\right)$
• D. $\left(- 6 , 4\right)$

Answer 1

Answer: (B)

Equation of family of circles touching line
$x+y=2$ at point $\left(1 , 1\right)$ is $\left(x - 1\right)^{2}+\left(y - 1\right)^{2}+\lambda \left(x + y - 2\right)=0$
Equation of common chord $PQ$ with the circle
$x^{2}+y^{2}+4x+5y-6=0$ is
$\left(- 6 x - 7 y + 8\right)+\lambda \left(x + y - 2\right)=0,$ which passes through the point $\left(6 , - 4\right)$ .