Let $(x, y)$ be any point on the parabola $y^2 = 4x$. Let $P$ be the point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1:3$. Then the locus of $P$ is

Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point $R$ lying on the line $y=2x+a$,where $a > 0$.
$1.$ The length of the chord $PQ$ is:
$(A)$ $7a$ $(B)$ $5a$ $(C)$ $2a$ $(D)$ $3a$
$2.$ If the chord $PQ$ subtends an angle $\theta$ at the vertex of the parabola $y^2=4ax$,then $\tan \theta$ is:
$(A)$ $\frac{2}{3}\sqrt{7}$ $(B)$ $\frac{-2}{3}\sqrt{7}$ $(C)$ $\frac{2}{3}\sqrt{5}$ $(D)$ $\frac{-2}{3}\sqrt{5}$

What is the common tangent to the parabola $y^{2} = 8ax$ and the circle $x^{2} + y^{2} = 2a^{2}$?

If the tangent to the parabola $y^2 = ax$ makes an angle of $45^{\circ}$ with the $x$-axis,then the point of contact is

The equation of the parabola whose focus is $(5, 3)$ and directrix is $3x - 4y + 1 = 0$ is:

If the normal drawn at $P(8, 16)$ to the parabola $y^2 = 32x$ meets the parabola again at $Q$,then the equation of the tangent drawn at $Q$ to the parabola is