Let $L$ is distance between two parallel normals of  , $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,\,\,\,a > b$ then maximum value of $L$ is

If the chord through the point whose eccentric angles are $\theta \,\& \,\phi $ on the ellipse,$(x^2/a^2) + (y^2/b^2) = 1$  passes through the focus, then the value of $ (1 + e)$ $\tan(\theta /2) \tan(\phi /2)$ is

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the point $A_1$ and $A_2$, respectively and the tangent $T_2$ touches $P$ and $E$ at the points $A_4$ and $A_3$, respectively. Then which of the following statements is(are) true?

($A$) The area of the quadrilateral $A_1 A _2  A _3 A _4$ is $35$ square units

($B$) The area of the quadrilateral $A_1 A_2 A_3 A_4$ is $36$ square units

($C$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-3,0)$

($D$) The tangents $T_1$ and $T_2$ meet the $x$-axis at the point $(-6,0)$

The length of the latus rectum of an ellipse is $\frac{1}{3}$ of the major axis. Its eccentricity is

If the tangent to the parabola $y^2 = x$ at a point $\left( {\alpha ,\beta } \right)\,,\,\left( {\beta  > 0} \right)$ is also a tangent to the ellipse, $x^2 + 2y^2 = 1$, then $a$ is equal to