A geometry problem by A Former Brilliant Member

Geometry Level 4

Given that the equation of blue ellipse is $\dfrac {x^{2}}{36} + \dfrac {y^{2}}{9} = 1$ and the equations of the two red circles are $(|x| - 3)^{2} + y^{2} = 9$ , find the Maximum possible area of the green circle that can be embedded within the blue ellipse and the two red circles as shown in the figure below.

$Note:$ This diagram given is not drawn to scale.

The answer is 1.76714.

1 solution

Otto Bretscher
Nov 8, 2018

Let $r$ be the radius of the green circle. Applying the Pythagorean theorem to the triangle with its vertices at the center of the green circle, the meeting point of the two red circles, and the center of one of the red circles, we find that $(3-r)^2+3^2=(3+r)^2$ , or $r=\frac{3}{4}$ . Thus the area of the green circle is $\pi r^2=\frac{9\pi}{16}\approx \boxed{1.767}$ .

That's a nice and simple way to solve this. Thanks!

A Former Brilliant Member - 2 years, 7 months ago

A geometry problem by A Former Brilliant Member

The answer is 1.76714.

1 solution

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