Find the length of the common chord of the ellipse $\frac{(x - 2)^2}{9} + \frac{(y + 2)^2}{4} = 1$ and the circle $x^2 + y^2 - 4x + 2y + 4 = 0$.

$A$ line perpendicular to the $X$-axis cuts the circle $x^2+y^2=9$ at $A$ and the ellipse $4x^2+9y^2=36$ at $B$ such that $A$ and $B$ lie in the same quadrant. If $\theta$ is the greatest acute angle between the tangents drawn to the curves at $A$ and $B$,then $\tan \theta=$

Let $PQ$ be a focal chord of the parabola $y^{2}=4x$ such that it subtends an angle of $\frac{\pi}{2}$ at the point $(3, 0)$. Let the line segment $PQ$ be also a focal chord of the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}$. If $e$ is the eccentricity of the ellipse $E$,then the value of $\frac{1}{e^{2}}$ is equal to

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 $P: y^2=12x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ 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 angle between the curves $2x^2 + y^2 = 20$ and $4y^2 - x^2 = 8$ at a point where they intersect in the $4^{th}$ quadrant is

If the tangent drawn to the parabola $y^2=4x$ at $(t^2, 2t)$ is the normal to the ellipse $4x^2+5y^2=20$ at $(\sqrt{5} \cos \theta, 2 \sin \theta)$,then