Question

The length of transverse common tangent to two circles is $5\text{units}$ and a direct common tangent is $15\text{units}$, then the product of the radii of two circles is

• A. $25$
• B. $50$
• C. $75$
• D. $30$

Answer 1

The correct option is B $50$

Length of direct common tangent
$\Rightarrow \sqrt{d^2-(r_1-r_2{)}^2}=15$
$\Rightarrow d^2-(r_1-r_2{)}^2={15}^2\cdots (i)$
Length of transverse common tangent
$\Rightarrow \sqrt{d^2-(r_1+r_2{)}^2}=5$
$\Rightarrow d^2-(r_1+r_2{)}^2=5^2\cdots (ii)$
Now, $\left(i\right)-\left(ii\right)$
$\Rightarrow (r_1+r_2{)}^2-(r_1-r_2{)}^2=200$
$\therefore r_1{r}_2=50$