Properties of Parabola

If $PSQ$ is the focal chord of the parabola $y^2-8x=2y-17$, such that $SP=6$. Then the length of $SQ$ is (where $S$ is focus)

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

The locus of the foot of perpendicular drawn from focus upon a variable tangent to the parabola ${\left(2x-y+1\right)}^2=\frac{8}{\sqrt{5}}\left(x+2y+3\right)$ is

If tangents are drawn from a point $P$ to the parabola $y^2=4x$ with focus $\left(S\right)$ such that the lengths of the tangents are $\sqrt{5}$ and $\sqrt{10},$ then $SP$ is equal to

If ${\ell }_1$ and ${\ell }_2$ are lengths of two perpendicular focal chord of parabola $y^2=16(x+1),$ then harmonic mean of ${\ell }_1$ and ${\ell }_2$ is

If the tangents at two points $\left(1, 2\right)$ and $\left(3, 6\right)$ on a parabola intersect at the point $\left(1, 6\right),$ then equation of directrix 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

A ray of light travelling along the line $y=7$ strikes the surface of the curve $y^2-8x=6y-17$, then the equation of line along which the reflected ray travels, is

Given the curve $x=a\left(\mathrm{c}\mathrm{o}\mathrm{s}\theta +\frac{\pi }{2}-\theta \right)$ , $y=a(1-\sin \mathrm{\theta }), a>0$. If the length of the sub-tangent to the curve at $\mathrm{\theta }=0$ is $L$, the value of $L$ is :

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

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

The line $x+2 y-1=0$ is a tangent to a parabola at point $A$, intersects the directrix at $\mathrm{B}$ and the tangent at vertex at $C$. Focus of parabola is $S(2,0)$ then find the value of $64\mathrm{AC}$. $\mathrm{BC}$

The figure shows a parabola with focus $F$. The equation of the tangent at a point $A$ on the parabola is given by $y=x$. If the projection of point $A$ on the directrix is $B$, $B=(-1,2)$ and $F=(p,q)$, find $p-q$.

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

Chord joining two distinct points $P\left(a,4b\right)$ and $Q\left(c,-\frac{16}{b}\right)$ (both are variable points) on the parabola $y^2=16x$ always passes through a fixed point $\left(\alpha ,\beta \right).$ Then, which of the following statements is correct?

Let $\mathrm{P}\mathrm{Q}$ be the focal chord of the parabola ${\mathrm{y}}^2=4\mathrm{x.}$ If the centre of the circle having $\mathrm{P}\mathrm{Q}$ as its diameter lies on the line $\mathrm{y}=\frac{-4}{\sqrt{5}}$ and the length of chord $\mathrm{P}\mathrm{Q}$ is $L$ units then $5L=$

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