Question

Two circles each of radius 5 units touch each other at the point $(1,2).$ If the equation of their common tangent is $4x+3y=10$, and $C_1(\alpha ,\beta )$ and $C_2(\gamma ,\delta )$ $C_1\ne C_2$ are their centres, then $|(\alpha +\beta )(\gamma +\delta )|$ is equal to _____

Answer 1

Answer: 40

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