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

$M_{C_1{C}_2}=\frac{3}{4}\Rightarrow \cos \theta =\frac{4}{5},\sin \theta =\frac{3}{5}$
Point $\frac{x-1}{\frac{4}{5}}=\frac{y-2}{\frac{3}{5}}=\pm 5\text{(by parametric form of line)}$
$\Rightarrow x-1=\pm 4\text{or}y-2=\pm 3$
$\Rightarrow x=5,y=5\text{or}x=–3,y=–1$
$C_1(5,5)\text{and}{C}_2(–3,–1)$
$\Rightarrow |(\alpha +\beta )(\gamma +\delta )|=|(5+5)(–3–1)|=40$