The coordinates of the ends of a focal chord of a parabola $y^2 = 4ax$ are $(x_1, y_1)$ and $(x_2, y_2)$. Then $x_1x_2 + y_1y_2$ has the value equal to

If the equation of the parabola with vertex $V \left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0$,then $\alpha + \beta + \gamma$ is equal to:

The equation of the parabola whose vertex is at $(2, -1)$ and focus at $(2, -3)$ is

What are the equations of the tangents at the endpoints of the latus rectum of the parabola $y^2 = 4ax$?

Find the parameter $t$ for the point $(4, -6)$ on the parabola $y^2 = 9x$.

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:
$(A)$ $\frac{2}{3}\sqrt{7}$ $(B)$ $\frac{-2}{3}\sqrt{7}$ $(C)$ $\frac{2}{3}\sqrt{5}$ $(D)$ $\frac{-2}{3}\sqrt{5}$