Two circles have a common external tangent of length 17 cm and a common internal tangent of length 9 cm. Find the product of the radii of the circles.

Solution

Solution: Let the radii be r₁ and r₂, and the distance between the centres be d. For the external tangent, \(17^2=d^2-(r_1-r_2)^2\). For the internal tangent, \(9^2=d^2-(r_1+r_2)^2\). Subtracting the equations gives \(17^2-9^2=(r_1+r_2)^2-(r_1-r_2)^2=4r_1r_2\). Thus, \(289-81=4r_1r_2\Rightarrow208=4r_1r_2\Rightarrow r_1r_2=52\). Trick: The difference between the squares of the external and internal tangent lengths equals four times the product of the radii.