Let $O$ be the vertex of the parabola $y^2 = 4x$ and its chords $OP$ and $OQ$ are perpendicular to each other. If the locus of the mid-point of the line segment $PQ$ is a conic $C$,then the length of its latus rectum is:

Three points $O(0,0)$,$P(a, a^2)$,and $Q(-b, b^2)$ with $a > 0$ and $b > 0$ lie on the parabola $y = x^2$. Let $S_1$ be the area of the region bounded by the line $PQ$ and the parabola,and $S_2$ be the area of the triangle $OPQ$. If the minimum value of $\frac{S_1}{S_2}$ is $\frac{m}{n}$,where $\operatorname{gcd}(m, n) = 1$,then $m + n$ is equal to:

Find the length of the latus rectum of the parabola $y = x \tan \alpha - \frac{g x^2}{2 u^2 \cos^2 \alpha}$.

The equation of the axis of the parabola $x^2 - 4x - 3y + 10 = 0$ is:

The line $y = mx + 1$ is a tangent to the parabola $y^2 = 4x$,if

$P$ and $Q$ are the extremities of a focal chord of the parabola $y^2=4ax$. If $P=(9,9)$ and $Q=(p, q)$,then $p-q=$