The values of $c$ such that the line $y=4x+c$ touches the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ are

The length of the latus rectum of the ellipse $16x^2 + 25y^2 = 400$ is

The foci of the ellipse $25x^2 + 4y^2 + 100x - 4y + 100 = 0$ are

$a$ and $b$ are the semi-major and semi-minor axes of an ellipse whose axes are along the coordinate axes. If its latus rectum is of length $4$ units and the distance between its foci is $4 \sqrt{2}$,then $a^2+b^2=$

The area of the quadrilateral formed by drawing tangents at the ends of the latus recta of the ellipse $\frac{x^2}{4} + \frac{y^2}{1} = 1$ is

The equation $2x^2 + 3y^2 - 8x - 18y + 35 = k$ represents: