$A$ tangent to the curve $x = a t^{2}, y = 2 a t$ is perpendicular to the $X$-axis. Then the point of contact is:

$A$ point on the parabola whose focus and vertex are respectively at $\left(\frac{5}{4}, -2\right)$ and $(1, -2)$ 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 length of the chord of the parabola $x^2 = 4ay$ passing through its vertex and having slope $\tan\alpha$ is:

The equations of the sides $AB$ and $AC$ of a triangle $ABC$ are $(\lambda+1) x +\lambda y =4$ and $\lambda x +(1-\lambda) y +\lambda=0$ respectively. Its vertex $A$ is on the $y$-axis and its orthocentre is $(1,2)$. The length of the tangent from the point $C$ to the part of the parabola $y^2=6 x$ in the first quadrant is

The axis of the parabola $9y^2 - 16x - 12y - 57 = 0$ is