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An ellipse and a hyperbola have their principal axes along the coordinate axes and have a common foci separated by a distance $2\sqrt{13}$,the difference of their focal semi axes is equal to $4$.If the ratio of their eccentricities is $\frac{3}{7}$.Find the equation of these curves.


The distance between the foci of both ellipse and hyperbola is $2\sqrt{13}$.

So $2c_1=$distance between the foci of ellipse$=2\sqrt{13}$

$2c_2=$distance between the foci of hyperbola$=2\sqrt{13}$

$\frac{\text{eccentricity of ellipse}}{\text{eccentricity of hyperbola}}=\frac{3}{7}$

I do not understand what is the meaning of "the difference of their focal semi axes" and how to proceed further.

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Focal semi axis is equal to the length of the semi axis passing through focus of ellipse or hyperbole. Focal semi axis mean same as semi major axis. I hope you can now get answers as: For ellipse: $a^2=49$ and $b^2=36$. For hyperbola: $a^2=9$ and $b^2=4$ Hope it helps.

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