An ellipse has its center at $(1, -2)$,one focus at $(3, -2)$,and one vertex at $(5, -2)$. Then the length of its latus rectum is :

If a tangent having slope $m = \frac{1}{3}$ to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ is a normal to the circle $(x + 1)^2 + (y + 1)^2 = 1$,then $a^2$ lies in the interval:

Which of the following statements are true and which are false? In each case,give a valid reason for your answer.
$r:$ $A$ circle is a particular case of an ellipse.

What is the equation of the chord of the ellipse $\frac{x^2}{36} + \frac{y^2}{9} = 1$ that is bisected at the point $(2, 1)$?

If the line $x \cos \alpha + y \sin \alpha = p$ is normal to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$,then

Let $E_1 = \frac{x^2}{9} + \frac{y^2}{4} = 1$ and $E_2 = \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be two ellipses and $R$ be a rectangle with sides parallel to the coordinate axes. Let $E_1$ be the inscribed ellipse in $R$ and $E_2$ be the circumscribed ellipse on $R$. If $E_2$ passes through $(0, 4)$,then: