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

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 chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1, \frac{2}{5}\right)$, is equal to:

In an ellipse, the distance between its foci is $6$ and minor axis is $8$. Then its eccentricity is

If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse $x^2+4 y^2=36$ is $r$, then $12 r^2$ is equal to

If the length of the major axis of an ellipse is three times the length of its minor axis, then its eccentricity is