Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(b < a)$, be a ellipse with major axis $A B$ and minor axis $C D$. Let $F_1$ and $F_2$ be its two foci, with $A, F_1, F_2, B$ in that order on the segment $A B$. Suppose $\angle F_1 C B=90^{\circ}$. The eccentricity of the ellipse is

The centre of an ellipse is $C$ and $PN$ is any ordinate and $A$, $A’$ are the end points of major axis, then the value of $\frac{{P{N^2}}}{{AN\;.\;A'N}}$ is

The distance between the directrices of the ellipse $\frac{{{x^2}}}{{36}} + \frac{{{y^2}}}{{20}} = 1$ is

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

If the points of intersection of two distinct conics $x^2+y^2=4 b$ and $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ lie on the curve $y^2=3 x^2$, then $3 \sqrt{3}$ times the area of the rectangle formed by the intersection points is............................

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