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 reciprocal of the square of the distance between the centres of the circles is equal to

Answer 1

Answer: 0.1

Let two circles are of radii $r_1$ & $r_2$ and with centres $C_1$ & $C_2$ respectively, then
${\left(C_1{C}_2\right)}^2-{\left(r_1-r_2\right)}^2=9$ &
${\left(C_1{C}_2\right)}^2-{\left(r_1+r_2\right)}^2=1$
$\Rightarrow 4r_1{r}_2=8\Rightarrow r_1{r}_2=2$
Assume, $r_1=2$ & $r_2=1$
Then, ${\left(C_1{C}_2\right)}^2=1+3^2=10$
$\frac{1}{{\left(C_1{C}_2\right)}^2}=0.1$