The locus of the midpoints of the chords of the hyperbola $x^2 - y^2 = a^2$ which touch the parabola $y^2 = 4ax$ is:

The circle $x^2 + y^2 = 5$ meets the parabola $y^2 = 4x$ at $P$ and $Q$. Then the length $PQ$ 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 $(-6,0)$.

What is the equation of the common tangent to the circle $(x - 3)^2 + y^2 = 9$ and the parabola $y^2 = 4x$ above the $X$-axis?

An equation for the line that passes through $(10, -1)$ and is perpendicular to $y = \frac{x^2}{4} - 2$ is

The line $y=x+5$ touches