Consider the ellipse $\frac{x^2}{\cos ^2 \alpha}+\frac{y^2}{\sin ^2 \alpha}=1$, where $\alpha \in\left(0, \frac{\pi}{4}\right)$. Then, locus of point of intersection of one of the directrix and tangent at upper end of minor axis is

If a man running around a race-course notes that the sum of the distances of two flag-posts from him is always $$10 \mathrm{~m}$$ and the distance between the flag-posts is $$8 \mathrm{~m}$$, then the area of the path he encloses in square metres is

The focal distance of the point $$(x, y)$$ from the ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$$ is

The eccentricity of the ellipse $$25 x^2+9 y^2-150 x-90 y+225=0$$ is