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  4. 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

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Q. 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

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Solution:

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, $\left(C_{1} C_{2}\right)^{2}=1+3^{2}=10$
$\frac{1}{\left(C_{1} C_{2}\right)^{2}}=0.1$