How to write a system of equation that satisfies the conditions "a circle and an ellipse that intersect at four points?"

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Answer 1 · 2016-10-27T20:05:17

Please see the explanation.

An ellipse with an equation:

#x^2/a^2 + y^2/b^2 = 1#

Has vertices at #(-a, 0), (a, 0), (0, -b), and (0, b)#

A circle has with an equation:

#x^2 + y^2 = r^2#

Has points #(-r, 0), (r,0),(0,-r), and (0, r)#

If #a > r > b#, then the circle is sure to intersect the ellipse at 4 points.

For example, #4 > 3 > 2# meets these conditions.

The corresponding equations will be:

#x^2/4^2 + y^2/2^2 = 1#

#x^2 + y^2 = 3^2#