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:

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ $x^2/a^2 + y^2/b^2 = 1$

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

A circle has with an equation:

$x^{2} + y^{2} = r^{2}$ $x^2 + y^2 = r^2$

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

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

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

The corresponding equations will be:

$\frac{x^{2}}{4^{2}} + \frac{y^{2}}{2^{2}} = 1$ $x^2/4^2 + y^2/2^2 = 1$

$x^{2} + y^{2} = 3^{2}$ $x^2 + y^2 = 3^2$