Question

Two circles each of radius 5 units touch each other at the point $(1,2) .$ If the equation of their common tangent is $4 x+3 y=10$, and $C_{1}(\alpha, \beta)$ and $C_{2}(\gamma, \delta)$ $C _{1} \neq C _{2}$ are their centres, then $|(\alpha+\beta)(\gamma+\delta)|$ is equal to _____

Answer 1

Slope of line joining centres of circles $=\frac{4}{3}=\tan \theta$

$\Rightarrow \cos \theta=\frac{3}{5}, \sin \theta=\frac{4}{5}$
Now using parametric form
$\frac{x-1}{\cos \theta}=\frac{y-2}{\sin \theta}=\pm 5 $
$\oplus (x, y)=(1+5 \cos \theta, 2+5 \sin \theta)$
$(\alpha, \beta)=(4,6)$
$\ominus ( x , y )=(\gamma, \delta)=(1-5 \cos \theta, 2-5 \sin \theta)$
$(\gamma, s )=(-2,-2) $
$\Rightarrow |(\alpha+\beta)(\gamma+\delta)|=|10 x -4|=40$