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
$\left(-6x-7y+8\right)+\lambda \left(x+y-2\right)=0,$ which passes through the point $\left(6,-4\right)$ .