Parabola practice · Mixed difficulty

Let $x = 2t,\,\,y = \frac{{{t^2}}}{3}$ be a conic. Let $S$ be the focus and $B$ be the point on the axis of the conic such that $SA \bot BA$ where $A$ is any point on the conic. If $k$ is the ordinate of the centroid of the $\Delta SAB,$ then $\lim\limits_{t \to 1} k\,$ is equal to

Find length of focal chord of parabola $y^2=8 x$ which is perpendicular to line $x+y=1$

The focus of the parabola $(y-2)^2=20(x+3)$ is

The number of distinct normals that can be drawn from $(-2,1)$ to the parabola $y^2-4 x-2 y-3=0$, is

If $x^2+6 x+20 y-51=0$, then the axis of parabola is

The centre of the conic $14 x^2-4 x y+11 y^2-44 x-58 y+71=0,$ is

If the line $y=m x+c$ is a tangent to the parabola $y^2=4 a(x+a)$ then $m a+\frac{a}{m}$ is equal to

The equation of a line touching both the parabolas $y^2=4 x$ and $x^2=-32 y$, is

The coordinates of the extremities of the latus rectum of the parabola $5 y^2=4 x$ are

If the common tangent to the parabolas $y^2=4 x$ and $x^2=4 y$ also touches the circle $x^2+y^2=c^2$, then $c$ is equal to

The area of the triangle inscribed in the parabola ${y^2} = 4x$ the ordinates of whose vertices are 1,2 and 4 is

The equation of the parabola whose vertex and focus are on the positive side of the $x$-axis at distances $a$ and $b$ respectively from the origin, is

If $(2,0)$ is the vertex and $y$-axis the directrix of a parabola, then its focus is

$PQ$ is any focal chord of the parabola $y^2=32 x$. The length of $P Q$ can never be less than

A triangle $ABC$ of area $\Delta$ is inscribed in the parabola ${y^2} = 4ax$ such that $A$ is the vertex and $BC$ is a focal chord of the parabola. The difference of the ordinates of $B$ and $C$ is

The point on $y^2=4 a x$ nearest to the focus has its abscissa equal to

The equation of the parabola whose vertex is $(-1,-2)$, axis is vertical, and which passes through the point $(3,6)$, is

If the line $x + y - 1 = 0$ is a tangent to the parabola ${y^2} - y + x = 0$, then the point of contact is

The equation of a common tangent to $y^2=4 x$ and the curve $x^2+4 y^2=8$ can be

Two perpendicular tangents ${PA}$ and ${PB}$ are drawn to ${y}^2=4 {ax}$, minimum length of ${AB}$ is equal to

The ends of a line segment are $P\left( {1,\;3} \right)$ and $Q\left( {1,\;1} \right).$ $R$ is a point on the line segment $PQ$ such that $PR:QR = 1:\lambda$. If $R$ is an interior point of the parabola ${y^2} = 4x$, then

The equation of a tangent to the parabola $y^2=4 a x$ which is making an angle $\theta$ with $x$-axis is

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

If $x = my + c$ is a normal to the parabola ${x^2} = 4\;ay,$ then the value of $c$ is

A line passing through $(21,30)$ and normal to the curve $y=2 \sqrt{x}$. If $\mathrm{m}$ is slope of the normal then $m+6=$

The length of the normal of the parabola ${y^2} = 4x$ which subtends a right angle at the vertex is

The equation of the common tangent to the parabolas $y^2=4ax$ and $x^2=4 b y$, is

Set of values of m for which a chord of slope m of the circle ${x^2} + {y^2} = 4$ touches the parabola ${y^2} = 4x,$ is

The line $x+y=6$ is a normal to the parabola $y^2=8 x$ at the point.

Consider the parabola $y^2=8 x$. Let $\Delta_1$ be the area of the triangle formed by the end points of its latus rectum and the point $P(1 / 2,2)$ on the parabola, and $\Delta_2$ be the area of the triangle formed by drawing tangents at $P$ and at the end-points of the latus rectum. Then, $\frac{\Delta_1}{\Delta_2}$, is