If the foci and vertices of an ellipse are $(\pm 1, 0)$ and $(\pm 2, 0)$ respectively,then the length of the minor axis of the ellipse is:

$P$ is a variable point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with foci $F_1$ and $F_2$. If $A$ is the area of the triangle $P F_1 F_2$,then the maximum value of $A$ is

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?

If the line joining the points $A(\alpha)$ and $B(\beta)$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ is a focal chord,then one possible value of $\cot \frac{\alpha}{2} \cdot \cot \frac{\beta}{2}$ is

Statement $I$: The equation of the directrix of the ellipse $4x^2+y^2-8x-4y+4=0$ is $3y=6-4\sqrt{3}$.
Statement $II$: The equation of the latus rectum of the ellipse $x^2+4y^2-4x-8y+4=0$ is $y=2+\sqrt{3}$.
Which of the above statement$(s)$ is (are) true?

The angle between the pair of tangents drawn from the point $(1, 2)$ to the ellipse $3x^2 + 2y^2 = 5$ is: