If the line $y = 4x + c$ is a tangent to the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$,then $c = \dots$

If the latus rectum of an ellipse is equal to half of its minor axis,then its eccentricity is

The eccentricity of the ellipse represented by the equation $7x^2 + 16y^2 - 14x + 64y - 377 = 0$ is...

$A$ tangent drawn at a point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ cuts the $X$-axis at point $A$. If $A^{\prime}$ is the image of $A$ with respect to the line $y=x$,then the circle with $AA^{\prime}$ as its diameter passes through the fixed point:

The equations of the tangents to the ellipse $9x^2 + 16y^2 = 144$ which pass through the point $(2, 3)$ are

$A$ staircase of length $l$ rests against a vertical wall and a floor of a room. Let $P$ be a point on the staircase,nearer to its end on the wall,that divides its length in the ratio $1 : 2$. If the staircase begins to slide on the floor,then the locus of $P$ is