How to write a system of equation that satisfies the conditions "a hyperbola and an ellipse that intersect in two points?"

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Answer 1 · 2017-05-04T02:10:34

Please see the explanation for 2 general examples.

A hyperbola with a horizontal transverse axis:

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{b^{2}} = 1$ $(x-h)^2/a^2-(y-k)^2/b^2=1$

is tangent at its 2 vertices to the 2 ends of major axis of a parabola:

$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1; a > b$ $(x-h)^2/a^2+(y-k)^2/b^2=1;a>b$

Where the values of a, h and k are the same for both equations.

A hyperbola with a vertical transverse axis:

$\frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{b^{2}} = 1$ $(y-k)^2/a^2-(x-h)^2/b^2=1$

is tangent at its 2 vertices to the 2 ends of major axis of a parabola:

$\frac{{(y - k)}^{2}}{a^{2}} + \frac{{(x - h)}^{2}}{b^{2}} = 1; a > b$ $(y-k)^2/a^2+(x-h)^2/b^2=1;a>b$

Where the values of a, h, and k are the same for both equations.