$A$ point moves such that the sum of its distances from $(ae, 0)$ and $(-ae, 0)$ is $2a$. Then the equation to its locus,where $b^2 = a^2(1 - e^2)$,is

If the tangent at a point on the ellipse $\frac{x^2}{27} + \frac{y^2}{3} = 1$ meets the coordinate axes at $A$ and $B,$ and $O$ is the origin,then the minimum area (in sq. units) of the triangle $OAB$ is

Find the area of the triangle formed by the $X$-axis and the tangent and the normal to the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\left(\frac{a}{\sqrt{2}}, \frac{b}{\sqrt{2}}\right)$.

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

Point $O$ is the centre of the ellipse with major axis $AB$ and minor axis $CD$. Point $F$ is one focus of the ellipse. If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is $2$,then the product $(AB)(CD)$ is equal to

Assertion $(A)$: The length of the latus rectum of an ellipse is $4$. The focus and its corresponding directrix are respectively $(1, -2)$ and $3x + 4y - 15 = 0$. Then its eccentricity is $\frac{1}{2}$.
Reason $(R)$: The length of the perpendicular drawn from the focus of an ellipse to its corresponding directrix is $\frac{a(1 - e^2)}{e}$.
Which one of the following is correct?