The normal at the point $(bt_1^2, 2bt_1)$ on the parabola $y^2 = 4bx$ meets the parabola again at the point $(bt_2^2, 2bt_2)$. Then:

The nearest point on the curve $x^2=2y$ to the point $(0,5)$ is . . . . . . .

The angle of intersection between the curves $x^2 = 8y$ and $y^2 = 8x$ at the origin is

Find the parametric coordinates of any point on the parabola whose focus is $(0, 1)$ and directrix is $x + 2 = 0$.

Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^{2}=4x$ be the curve $S$. Let $P$ be any point on $S$. Then the locus of the point, which internally divides $OP$ in the ratio $3:1$, is:

Let $PQ$ be a focal chord of the parabola $y^2=36x$ of length $100$,making an acute angle with the positive $x$-axis. Let the ordinate of $P$ be positive and $M$ be the point on the line segment $PQ$ such that $PM:MQ=3:1$. Then which of the following points does $NOT$ lie on the line passing through $M$ and perpendicular to the line $PQ$?