The difference of the focal semi axes of an ellipse and a hyperbola is equal to $4$.If the ratio of their eccentricities is $\frac{3}{7}$. 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.
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.
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.