Question

Two non congruent circles are externally tangent. The product of their radii is an integer $k$ between $1$ and $100$ inclusive. Number of values of $k$ for which the length of an external tangent is also an integer, is

Answer 1

$L^{2}=\sqrt{d^{2}-\left(r_{2}-r_{1}\right)^{2}}$
$L=\sqrt{\left(r_{1}+r_{2}\right)^{2}-\left(r_{2}-r_{1}\right)^{2}}$

$L=\sqrt{4 r_{1} r_{2}}=2 \sqrt{r_{1} r_{2}}$
$L=2 \sqrt{k} \,\,\, k \in[1,100]$
$L$ will be an integer if
$k=1,4,9,16,25,36,49,64,81,100$
i.e. $10$ values