If the curves, $\frac{x^{2}}{a}+\frac{y^{2}}{b}=1$ and $\frac{x^{2}}{c}+\frac{y^{2}}{d}=1$ intersect each other at an angle of $90^{\circ},$ then which of the following relations is TRUE ?

The equation to the locus of the middle point of the portion of the tangent to the ellipse $\frac{{{x^2}}}{{16}}$$+$ $\frac{{{y^2}}}{9}$ $= 1$  included between the co-ordinate axes is the curve :

The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$. If one of the directrices is $x = 4$, then the equation of the ellipse is

Let the equations of two ellipses be ${E_1}:\,\frac{{{x^2}}}{3} + \frac{{{y^2}}}{2} = 1$ and ${E_2}:\,\frac{{{x^2}}}{16} + \frac{{{y^2}}}{b^2} = 1,$ If the product of their eccentricities is $\frac {1}{2},$ then the length of the minor axis of ellipse $E_2$ is

The ellipse $ 4x^2 + 9y^2 = 36$  and the hyperbola $ 4x^2 -y^2 = 4$  have the same foci and they intersect at right angles then the equation of the circle through the points of intersection of two conics is

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