If the line $y = mx + c$ touches the ellipse $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$,then $c = $

Let a focus of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be $S(4, 0)$ and its eccentricity be $\frac{4}{5}$. If the point $P(3, \alpha)$ lies on $E$ and $O$ is the origin, then the area of $\triangle POS$ is equal to: (in $/ 5$)

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

The value of $k$,if $(1, 2)$ and $(k, -1)$ are conjugate points with respect to the ellipse $2x^2 + 3y^2 = 6$,is

Find the locus of the midpoint of the portion of any tangent to the ellipse $x^{2} + 2y^{2} = 2$ intercepted between the axes.

The equation of the ellipse passing through the origin and having foci at $(1, 0)$ and $(3, 0)$ is .....