If the line $7y - 4x = 10$ is a tangent to the parabola $y^2 = 4x$,find the point of contact.

What is the area of the triangle formed by the tangents drawn from the point $(4, 6)$ to the parabola $y^2 = 8x$ and their chord of contact?

The focal distance of a point $P$ on the parabola $y^{2}=12x$ if the ordinate of $P$ is $6$,is

Find the coordinates of the focus,axis of the parabola,the equation of the directrix,and the length of the latus rectum for $x^{2}=6y$.

The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^{2}=8x$, with one of its vertices at the vertex of this parabola, is (in $\sqrt{3}$)

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 $TRUE$?
$(A)$ $SQ_1=2$
$(B)$ $Q_1Q_2=\frac{3\sqrt{10}}{5}$
$(C)$ $PQ_1=3$
$(D)$ $SQ_2=1$