Parabola PYQs - Last 10 Years

If the locus of a point that divides a chord of slope 2 of the parabola $y^2=4 x$ internally in the ratio $1: 2$ is a parabola, then its vertex is

A circle is drawn with its centre at the focus of the parabola $y^2=2 p x$ such that it touches the directrix of the parabola. Then, a point of intersection of the circle and the parabola is

If $x-y-3=0$ is a normal drawn through the point $(5,2)$ to the parabola $y^2=4 x$, then the slope of the other normal that can be drawn through the same point to the parabola $y^2=4 x$ is

If the normal chord drawn at the point $\left(\frac{15}{2}, \frac{15}{\sqrt{2}}\right)$ to the parabola $y^2=15 x$ subtends an angle $\theta$ at the vertex of the parabola, then $\sin \frac{\theta}{3}+\cos \frac{2 \theta}{3}-\sec \frac{4 \t...

Tangents are drawn at three points $P\left(t_1\right), Q\left(t_2\right), R\left(t_3\right)$ on the parabola $y^2=x$. Let these tangents intersect each other at the points $L, M, N$. If $t_1=2, t_2=-4, t_3=6$, then the area of the $\triangl...

The lengths of the two focal chords of the parabola $y^2=16 x$ is 25 units each. If these two chords cut the parabola at $A, B, C$ and $D$, then the area (in sq. units) of the quadrilateral formed by $A, B, C$ and $D$ is

If the tangents of the parabola $y^2=8 x$ passing through the point $P(1,3)$ touches the parabola at $A$ and $B$, then the area (in sq. units) of $\triangle P A B$ is

$P Q$ is a focal chord of the parabola $y^2=4 x$ with focus $S$. If $P=(4,4)$, then $S Q=$

If the perpendicular distance from the focus of a parabola $y^2=4 a x$ to its directrix is $\frac{3}{2}$, then the equation of the normal drawn at $(4 a,-4 a)$ is

The angle between the tangents drawn from the point $(1,4)$ to the parabola $y^2=4 x$ is

If $L$ is the normal drawn to the parabola $y^2=8 x$ at the point $t=\frac{1}{\sqrt{2}}$, then the foot of the perpendicular drawn from the focus of the parabola on to the normal $L$ is

If $P$ is a point which divides the line segment joining the focus of the parabola $y^2=12 x$ and a point on the parabola in the ratio $1: 2$. Then, the locus of $p$ is

If the normal chord drawn at $(2 a, 2 a \sqrt{2})$ on the parabola $y^2=4 a x$ subtends an angle $\theta$ at its vertex, then $\theta=$

Equation of the line touching both parabolas $y^2=4 x$ and $x^2=-32 y$ is

If the ordinates of points $P$ and $Q$ on the parabola $y^2=12 x$ are in the ratio $1: 2$. Then, the locus of the point of intersection of the normals to the parabola at $P$ and $Q$ is

A common tangent to the circle $x^2+y^2=9$ and parabola $y^2=8 x$ is

The normal drawn at a point $(2,-4)$ on the parabola $y^2 \pm 8 x$ cuts again the same parabola at $(\alpha, \beta)$, then $\alpha+\beta=$

The line $x-2 y-3=0$ cuts the parabola $y^2=4 \operatorname{ar}$ at the points $P$ and $Q$. If the focus of this parabola is $\left(\frac{1}{4}, k\right)$. then $P Q=$

If the axes are rotated through an angle $45^{\circ}$ about the origin in anticlockwise direction, then the transformed equation of $y^2=4 a r$ is

Let the equation of the tangent at a point $P$ on the parabola $x^2-4 x-4 y+16=0$ be $2 x-y-5=0$. If the equation of the normal drawn at $P$ to this parabola is $a x+y+c=0$, then $a c=$

The perpendicular distance from the origin to the focal chord drawn through the point $(4,5)$ to the parabola $y^2-4 y-3 x+7=0$ is

Which of the following represents a parabola?

If \(a x+b y=1\) is a normal to the parabola \(y^2=4 p x\), then the condition is

Suppose a parabola with focus at \((0,0)\) has \(x-y+1=0\) as its tangent at the vertex. Then, the equation of its directrix is

Suppose a parabola passes through \((0,4),(1,9)\) and \((4,5)\) and has its axis parallel to the \(Y\)-axis. Then, the equation of the parabola is

The coordinates of the focus of the parabola described parametrically by \(x=5t^2+2\) and \(y=10t+4\) (where t is a parameter) are

The point of intersection of the latus rectum and axis of the parabola \(y^2+4 x+2 y-8=0\) is

If one end of focal chord of the parabola \(y^2=8x\) is \(\left(\frac{1}{2},2\right)\), then the length of the focal chord is ................ units.

Find the equation of the parabola which
passes through (6, \(-\)2), has its vertex at the
origin and its axis along the Y-axis.