Question

A circle $C$ having center at $\left(1,2\right)$ and radius equal to $3$ , cuts the members of the family of circles passing through two fixed points $P\left(2,6\right)$ and $Q\left(4,5\right)$ , such that the common chords pass through a fixed point $\left(x_1,y_1\right)$ , then the value of $\left(\frac{y_1-x_1}{7}\right)$ is

Answer 1

Answer: 4

Equation of family of circles:
$\left(x-2\right)\left(x-4\right)+\left(y-6\right)\left(y-5\right)+\lambda \left(x+2y-14\right)=0$
$x^2+y^2+\left(\lambda -6\right)x+\left(2\lambda -11\right)y+\left(38-14\lambda \right)=0$
Equation of circle ' $C$ ': $x^2+y^2-2x-4y-4=0$
Equation of common chords:
$\left(\lambda -4\right)x+\left(2\lambda -7\right)y+\left(42-14\lambda \right)=0$
$\lambda \left(x+2y-14\right)-\left(4x+7y-42\right)=0$
$\therefore$ Fixed point $\left(x_1,y_1\right)=\left(-14,14\right)$