The number of values of $c$ such that the line $y = cx + c$,where $c \in R$,touches the ellipse $\frac{x^2}{4} + \frac{y^2}{1} = 1$ is:

On the ellipse $4x^2 + 9y^2 = 1$,the points at which the tangents are parallel to the line $8x = 9y$ are

The locus of mid-points of the line segments joining $(-3,-5)$ and the points on the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ is :

The area of the quadrilateral formed by the tangents at the end points of the latus rectum to the ellipse $\frac{x^2}{9} + \frac{y^2}{5} = 1$ is .............. $sq. \text{ units}$.

If the origin is the centre,the $X$-axis is the major axis,and $\sqrt{\frac{2}{5}}$ is the eccentricity of an ellipse which passes through $(-3, 1)$,then the equation of that ellipse is:

Let $S$ and $S^{\prime}$ be the foci of an ellipse $E$ and $B$ be one end of its minor axis. Let $\angle S^{\prime} SB = \frac{\pi}{6}$ and $(2 \sqrt{3}, 1)$ be a point on $E$. If $X$-axis is the major axis and $Y$-axis is the minor axis of the ellipse $E$,then the sum of the squares of the lengths of major axis and minor axis is