${\text{ An ellipse intersects the hyperbola 2x}}^2-2y^2=1 \text{orthogonally. The eccentricity of the ellipse is reciprocal}$

$\text{of that of the hyperbola. If the axes of the ellipse are along the coordinates axes, then}$

The ellipse and hyperbola will be confocal.

$\text{ Foci of given hyperbola are }(\pm 1,0)\text{ . }$

$\text{ Let equation of ellipse be }\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\text{ . }$

$\begin{array}{l}\therefore \left(\pm a\times \frac{1}{\sqrt{2}},0\right)\equiv (\pm 1,0)\\ \text{ or }a=\sqrt{2}\text{ and }e=\frac{1}{\sqrt{2}}\\ \text{ or }{b}^2=a^2\left(1-e^2\right)\\ \text{ or }{b}^2=1\end{array}$

$\text{Therefore, the equation of the ellipse is}\frac{x^2}{2}+\frac{y^2}{1}=1$