Let $P$ be the point $(1, 0)$ and $Q$ be a point on the locus $y^2 = 8x$. The locus of the midpoint of $PQ$ is

$A$ parabola has the origin $(0,0)$ as its focus and the line $x = 2$ as the directrix. Then the vertex of the parabola is at

The point on the parabola $y^2 = 8x$ at which the normal is parallel to the line $x - 2y + 5 = 0$ is

Let $P(x, y)$ be the midpoint of the line joining $(1, 0)$ to a point on the curve $y^{2} = \left|\begin{array}{ll}x+1 & x+2 \\ x+3 & x+5\end{array}\right|$. Then,the locus of $P$ is symmetrical about

Suppose $AB$ is a focal chord of the parabola $y^2=12x$ of length $l$ and slope $m < \sqrt{3}$. If the distance of the chord $AB$ from the origin is $d$,then $l \cdot d^2$ is equal to ....................

Find the equation of the normal to the parabola $y^2 = 8x$ at the point $(2, 4)$.