The locus of point of intersection of two perpendicular tangent of the ellipse  $\frac{{{x^2}}}{{{9}}} + \frac{{{y^2}}}{{{4}}} = 1$ is :-

The equation of the ellipse referred to its axes as the axes of coordinates with latus rectum of length $4$ and distance between foci $4 \sqrt 2$ is-

The equation of the ellipse whose one focus is at $(4, 0)$ and whose eccentricity is $4/5$, 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

The foci of the ellipse $25{(x + 1)^2} + 9{(y + 2)^2} = 225$ are at

A wall is inclined to the floor at an angle of $135^{\circ}$. A ladder of length $l$ is resting on the wall. As the ladder slides down, its mid-point traces an arc of an ellipse. Then, the area of the ellipse is