The parametric equations of the parabola $y^2 - 12x - 2y - 11 = 0$ are:

If the chord joining the points $P_{1}(x_{1}, y_{1})$ and $P_{2}(x_{2}, y_{2})$ on the parabola $y^{2} = 12x$ subtends a right angle at the vertex of the parabola, then $x_{1}x_{2} - y_{1}y_{2}$ is equal to

Let a parabola $P$ be such that its vertex and focus lie on the positive $x$-axis at a distance $2$ and $4$ units from the origin,respectively. If tangents are drawn from $O(0,0)$ to the parabola $P$ which meet $P$ at $S$ and $R$,then the area (in $sq. \text{ units}$) of $\triangle SOR$ is equal to:

What is the equation of the chord of contact drawn from the point $(2, 4)$ to the parabola $y^2 = 4x$?

Let $PQ$ be a double ordinate of the parabola $y^2 = -4x$,where $P$ lies in the second quadrant. If $R$ divides $PQ$ in the ratio $2 : 1$,then the locus of $R$ is:

Let $P : y^{2} = 4ax, a > 0$ be a parabola with focus $S$. Let the tangents to the parabola $P$ that make an angle of $\frac{\pi}{4}$ with the line $y = 3x + 5$ touch the parabola $P$ at $A$ and $B$. Then the value of $a$ for which $A, B$ and $S$ are collinear is: