The area of the triangle formed by the lines joining the vertex of the parabola,$x^2 = 8y$,to the extremities of its latus rectum is

Consider the parabola $y^2=4x$. Let $S$ be the focus of the parabola. $A$ pair of tangents drawn to the parabola from the point $P=(-2,1)$ meet the parabola at $P_1$ and $P_2$. Let $Q_1$ and $Q_2$ be points on the lines $SP_1$ and $SP_2$ respectively such that $PQ_1$ is perpendicular to $SP_1$ and $PQ_2$ is perpendicular to $SP_2$. Then,which of the following is/are $TRUE$?
$(A)$ $SQ_1=2$
$(B)$ $Q_1Q_2=\frac{3\sqrt{10}}{5}$
$(C)$ $PQ_1=3$
$(D)$ $SQ_2=1$

If the line $x + my + am^2 = 0$ is tangent to the parabola $y^2 = 4ax$,find the point of contact.

The angle subtended by the normal chord at the point $(9, 9)$ on the parabola $y^2 = 9x$ at the focus of the parabola is (in $^{\circ}$)

Statement $I$: $4x^2+y^2-4xy-30x-50y+40=0$ is the equation of a parabola having $(2,3)$ as its focus and $x+2y+5=0$ as its directrix.
Statement $II$: The equation of the directrix of the parabola $x^2-4x+16y+52=0$ is $y+1=0$.
Which of the above statements is (are) true?

The equation of the diameter of the parabola $y^2 = x$ corresponding to the chord $x - y + 1 = 0$ is