The locus of the point of intersection of perpendicular tangents to the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ is:

An ellipse is drawn such that the diameter of the circle $(x - 1)^2 + y^2 = 1$ is the semi-minor axis and the diameter of the circle $x^2 + (y - 2)^2 = 4$ is the semi-major axis. If the center of the ellipse is at the origin and its axes are the coordinate axes,find the equation of the ellipse.

$A$ man running on a racecourse notes that the sum of the distances from the two flag posts to him is always $10 \, m$ and the distance between the flag posts is $8 \, m$. Find the equation of the path traced by the man.

The tangent to the ellipse $9x^2 + 16y^2 = 288$ making equal intercepts on the coordinate axes intersects the $X$-axis and the $Y$-axis at points $A$ and $B$ respectively. Then,the area of $\triangle OAB$ (where $O$ is the origin) is:

Let $E_1 = \frac{x^2}{9} + \frac{y^2}{4} = 1$ and $E_2 = \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be two ellipses and $R$ be a rectangle with sides parallel to the coordinate axes. Let $E_1$ be the inscribed ellipse in $R$ and $E_2$ be the circumscribed ellipse on $R$. If $E_2$ passes through $(0, 4)$,then:

The distance between the directrices of the ellipse $\frac{x^2}{36} + \frac{y^2}{20} = 1$ is