Coordinate Geometry: JEE Main 2022 Mathematics question

For the hyperbola $\mathrm{H}: x^{2}-y^{2}=1$ and the ellipse $\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$, a $>\mathrm{b}>0$, let the (1) eccentricity of $\mathrm{E}$ be reciprocal of the eccentricity of $\mathrm{H}$, and (2) the line $y=\sqrt{\frac{5}{2}} x+\mathrm{K}$ be a common tangent of $\mathrm{E}$ and $\mathrm{H}$. Then $4\left(\mathrm{a}^{2}+\mathrm{b}^{2}\right)$ is equal to _____________.