Question

The length of a common internal tangent to two circles is $7$ and a common external tangent is $11$. If the product of the radii of the two circles is $p$, then the value of $\frac{p}{2}$ is

Answer 1

We know that, the length of common internal tangent
$7=\sqrt{d^2-(r_1+r_2{)}^2}\Rightarrow (r_1+r_2{)}^2=d^2-49\cdots (1)$

We know that, the length of common external tangent
$11=\sqrt{d^2-(r_1-r_2{)}^2}\Rightarrow (r_1-r_2{)}^2=d^2-121\cdots (2)$

From equation $\left(1\right)$ and $\left(2\right)$, we get
$4r_1{r}_2=72\Rightarrow 4p=72\therefore \frac{p}{2}=9$