Question

Two circles of radii $aandb$ $(a>b)$ touch each other externally. $ST$ is a common tangent touching the circles at $S$ and $T$ respectively, then $S{T}^2$ is equal to

• A. $a+b$
• B. $ab$
• C. $2ab$
• D. $4ab$

Answer 1

The correct option is D $4ab$

Take a line parallel to AB with an end at T

$CT\parallel AB$

$AB=a+b$

$CS=a-b$

WKT $\angle AST=90$

In $△CST$

$C{S}^2+C{T}^2=S{T}^2$

$⟹S{T}^2=(a+b{)}^2+(a–b{)}^2=2ab–(-2ab)=4ab$