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

From a point $(d, 0)$, three normals are drawn to the parabola $y^{2} = x$. Then:

Two perpendicular tangents to the parabola $y^2 = 4ax$ always intersect on the line:

Let $A, B$ and $C$ be the vertices of a variable right-angled triangle inscribed in the parabola $y^2 = 16x$. Let the vertex containing the right angle be $C = (4, 8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_o$. Then three times the length of the latus rectum of $C_o$ is . . . . . .

$A$ ray of light moving parallel to the $x$-axis gets reflected from a parabolic mirror whose equation is $(y - 2)^2 = 4(x + 1)$. After reflection,the ray must pass through the point:

The equation of the tangent to the parabola $y^{2}=4x$ inclined at an angle of $\frac{\pi}{4}$ to the positive direction of $x$-axis is: