Question

If the length of direct common tangent and transverse common tangent of two circles with integral radii are $3$ units and $1$ unit respectively, then the square of the distance between the centres of the circles is equal to

Answer 1

Answer: 10

Let two circles are of radii $r_1$ & $r_2$ and with centres $C_1$ & $C_2$ respectively, then
${\left(C_1{C}_2\right)}^2-{\left(r_1-r_2\right)}^2=9$ &
${\left(C_1{C}_2\right)}^2-{\left(r_1+r_2\right)}^2=1$
$\Rightarrow 4r_1{r}_2=8\Rightarrow r_1{r}_2=2$
Since radii are integral, so, $r_1=2$ & $r_2=1$ or $r_1=1\&r_2=2$
Then, ${\left(C_1{C}_2\right)}^2=1+3^2=10$