If $0 \leq x \leq 5$,then the minimum distance from the point $(0, c)$ to the parabola $y = x^2$ is:

If the tangent at a point $P,$ with parameter $t,$ on the curve $x = 4t^2 + 3, y = 8t^3 - 1, t \in R,$ meets the curve again at a point $Q,$ then the coordinates of $Q$ are

Let $PQ$ and $RT$ be two focal chords of the parabola $y^2=16x$. If $P=(4,8)$ and $R=(16,16)$,then the length of $QT$ is:

The length of the latus rectum of a parabola whose directrix is $x + y - 2 = 0$ and focus is $(3, -4)$ 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:

The points of intersection of the parabolas $y^2=5x$ and $x^2=5y$ lie on the line