Critical Radius

Geometry Level 4

An ellipse has the equation

x 2 9 + y 2 4 = 1 \dfrac{x^2}{9} + \dfrac{y^2}{4} = 1

And a family of circles centered at ( 5 , 6 ) (5, 6) is given by

( x 5 ) 2 + ( y 6 ) 2 = r 2 (x - 5)^2 + (y - 6)^2 = r^2

For two critical values of r r the circle and the ellipse intersect at a single point. Find the sum of these two values of r r


The answer is 15.746.

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