Two circles have a common external tangent of length 9 and a common internal tangent of length 5. Find the product of the radii of the circles.

Solution

Solution: Let the radii be r₁ and r₂ and the distance between centres be d. For the external tangent, \(d^2-(r_1-r_2)^2=9^2=81\). For the internal tangent, \(d^2-(r_1+r_2)^2=5^2=25\). Subtracting gives \((r_1+r_2)^2-(r_1-r_2)^2=56\Rightarrow4r_1r_2=56\Rightarrow r_1r_2=14\). Trick: Subtract the squared tangent lengths to eliminate the distance between centres.