Question

An elliptical zen garden has major and minor axis lengths of 16 and 12 respectively. One of the foci, F, is taken and a water pond (in the shape of a circle) with radius as small as possible is drawn tangent to the garden with center F. What is the radius of the pond? I've got the equation of the ellipse which I believe is: \[\frac{x^{2}}{64} + \frac{y^{2}}{36} = 1\text{ or }\frac{x^{2}}{36} + \frac{y^{2}}{64} = 1\]

Answer 1

Although I don't think that it matters which way it is oriented since it isn't specified.

Answer 2

Do I just set up a system?
\[9x^{2} + 16y^{2} = 576\]\[(x - 2\sqrt{7})^{2} + y^{2} = r^{2}\]Is there an easier way to do this?

Answer 3

@powerangers69 \(2\sqrt{7}\) is the focal distance from the center of the ellipse to one of the foci. Not the shortest tangent radius of the circle.

Answer 4

@dpaInc

Answer 5

is the radius 8pm2sqrt(7)

Answer 6

is the problem saying the center of the circular pond is one of the foci of the ellipse?

Answer 7

Yes.

Answer 8

The answer I believe is supposed to be \(8 - 2\sqrt{7}\). Not completely sure because this test had a lot of disputes from the solutions, but that's what they gave me. Not sure how they got that.

Answer 9

:c