Let $$\mathrm{H}_{\mathrm{n}}: \frac{x^{2}}{1+n}-\frac{y^{2}}{3+n}=1, n \in N$$. Let $$\mathrm{k}$$ be the smallest even value of $$\mathrm{n}$$ such that the eccentricity of $$\mathrm{H}_{\mathrm{k}}$$ is a rational number. If $$l$$ is the length of the latus rectum of $$\mathrm{H}_{\mathrm{k}}$$, then $$21 l$$ is equal to ____________.

Let the eccentricity of an ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ is reciprocal to that of the hyperbola $$2 x^{2}-2 y^{2}=1$$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is ___________.

The vertices of a hyperbola H are ($$\pm$$ 6, 0) and its eccentricity is $${{\sqrt 5 } \over 2}$$. Let N be the normal to H at a point in the first quadrant and parallel to the line $$\sqrt 2 x + y = 2\sqrt 2 $$. If d is the length of the line segment of N between H and the y-axis then d$$^2$$ is equal to _____________.

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 _____________.