The lengths of the sides of the rectangle of greatest area that can be inscribed in the ellipse $x^2+4y^2=64$ are

Find the condition for the line $ax + by + c = 0$ to be a normal to an ellipse $\frac{x^2}{4} + \frac{y^2}{36} = 1$.

The point $(4, -3)$ with respect to the ellipse $4x^2 + 5y^2 = 1$ is:

The eccentricity of the ellipse $4x^2 + 9y^2 = 36$ is

Let the line $2x + 3y - k = 0, k > 0$,intersect the $x$-axis and $y$-axis at the points $A$ and $B$,respectively. If the equation of the circle having the line segment $AB$ as a diameter is $x^2 + y^2 - 3x - 2y = 0$ and the length of the latus rectum of the ellipse $x^2 + 9y^2 = k^2$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $2m + n$ is equal to

An ellipse is described by using an endless string which is passed over two pins. If the axes are $6 \ cm$ and $4 \ cm$,the necessary length of the string and the distance between the pins respectively in $cm$,are