Properties of Parabola

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

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

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

Suppose $OABC$ is a rectangle in the $xy$-plane where $O$ is the origin and $A,B$ lie on the parabola $y=x^2.$ Then $C$ must lie on the curve-

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)

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

Three distinct normals to the parabola $y^2=x$ are drawn through a point $\left(c, 0\right)$, then

The number of distinct normals that can be drawn to parabola $y^2=16x$ from the point $(2,0)$, is

Angle between the tangents drawn to $y^2=4x$ at the points where it is intersected by the line $y=x-1$ is equal to

Let a focal chord of parabola $y^2=16x$ cuts it at points $\left(f, g\right)$ and $\left(h, k\right)$. Then $fh$ is equal to

Let $Q$ be a point on the parabola $(x + 2{)}^2=4y$ such that the sum of its distance from focus and point $P(1, 2)$ is minimum. Then the coordinates of $Q$ are?

The mirror image of the directrix of the parabola $y^2=4(x+1)$ in the line mirror $x+2y=3,$ is

If $P(a{t}^2, 2at)$ be one end of a focal chord of the parabola $y^2=4ax$, then the length of the chord is

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

If $P(a{t}^2, 2at)$ be one end of a focal chord of the parabola $y^2=4ax$, then the length of the chord is

If $t_{1 }\mathrm{a}\mathrm{n}\mathrm{d} t_2$ be the parameters of the end points of a focal chord for the parabola $y^{2 }=4ax,$ then which one is true?

If $P(a{t}^2, 2at)$ be one end of a focal chord of the parabola $y^2=4ax$, then the length of the chord is

The latus rectum of the parabola $y^2=4ax$, whose focal chord $P_1{P}_2$ such that $S{P}_1=3$ and $S{P}_2=2,$ where $S$ is focus, then length of the latus rectum (in units) is: