Properties of Parabola

At a point $P$ on the parabola $y^2=4ax\left(a>0\right)$ tangent and normal are drawn. Tangent intersects the $x$-axis at $Q$ and normal intersects the parabola at $R$ such that chord $PR$ subtends $90°$ at its vertex. Then which of the following is/are TRUE?

Let $P$ and $Q$ be distinct points on the parabola ${\mathrm{y}}^2=2\mathrm{x}\mathrm{ }$ such that a circle with $PQ$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $\mathrm{\Delta }OPQ$ is $3\sqrt{2},\mathrm{ }$ then the coordinates of $P$ can be

A circle passes through the points of intersection of the parabola $\mathrm{y}+1=(\mathrm{x}-4{)}^2$ and $\mathrm{x}$-axis. Then the length of tangent from origin to the circle is

Let the focus $S$ of the parabola $y^2=8x$ lie on the focal chord $PQ$ of the same parabola. If the length $QS=3$ units, then the ratio of length $PQ$ to the length of the latus rectum of the parabola is

If one end of a focal chord of the parabola $y^2=4x$ is $\left(1,2\right)$, the other end lies on

Let $3x-y-8=0$ be the equation of tangent to a parabola at the point $\left(7,13\right)$. If the focus of the parabola is at $\left(-1,-1\right)$, then the equation of its directrix is

A variable chord $\mathrm{PQ}$ of ${\mathrm{y}}^2=4ax$  subtends a right angle at $(0, 0)$ then the locus of point intersection of normals at $P$ and $Q$ will be _____

A circle is drawn through the point of intersection of the parabola $y=x^2-5x+4$ and the $x$-axis

such that origin lies outside it . The length of a tangent to the circle from the origin is :

The length of focal chord to the parabola $y^2=12x$ drawn from the point $\left(3,6\right)$ on it is :

The sides $AB$ and $AC$ of a triangle $ABC$ are given in position and the harmonic mean between the lengths $AB$ and $AC$ is also given; prove that the locus of the parabola touching the sides at $B$ and $C$is a circle whose centre lies on the line bisecting the angle $BAC.$

The locus of the trisection point of any arbitrary double ordinate of the parabola $x^2=4y,$ is

Maximum number of points on parabola $y^2=16x$ which are equidistant from a variable point $P$ (which lie inside the parabola), is/are:

The coordinates of focus of a parabola which touches the lines $x=0, y=0, x+y=1 \& y=x-2$ are

If the line $y-\sqrt{3}x+3=0$ cuts the parabola $y^2=x+2$ at $A$ and $B$, then $PA\cdot PB$ is equal to where [where $P=\left(\sqrt{3}, 0\right)$].

If $a,b$ and $c$ form a geometric progression with common ratio $r,$ then the sum of the ordinates of the points of intersection of the line $ax+by+c=0$ and the curve $x+2y^2=0$ is

A line is drawn from $A\left(-2,0\right)$ to intersect the curve $y^2=4x$ in $P$ and $Q$ in the first quadrant such that $\frac{1}{AP}+\frac{1}{AQ}<\frac{1}{4}$ , then slope of the line is always -

The set of values of $a$ for which at least one tangent to the parabola $y^2=4ax$ becomes normal to the circle $x^2+y^2- 2ax- 4ay+3a^2=0$, is (where $a$ is a real number)

On the parabola $y=x^2$, the point at least distance from the straight line $y=2x-4$ is

The equation of the mirror that can reflect all incident rays from origin parallel to $y$-axis is

The normal to the parabola $y^2=8x$ at the point $\left(2,4\right)$ meets the parabola again at the point