Parabola — JEE Main PYQs

Axis of a parabola lies along $x$ - axis. If its vertex and focus are at distance 2 and 4 respectively from the origin, on the positive $x$ - axis then which of the following points does not lie on it?

Equation of a common tangent to the circle, $x^2+y^2-6x=0$ and the parabola, $y^2=4x$, is:

Tangent and normal are drawn at $\mathrm{P}(16,16)$ on the parabola $y^2=16x$, which intersect the axis of the parabola at A and B , respectively. If C is the centre of the circle through the points $\mathrm{P},\mathrm{A}$ and B and $\angle \mathrm{CPB}=\theta$, then …

Let P be the point on the parabola, ${\mathrm{y}}^2=8\mathrm{x}$ which is at a minimum distance from the centre C of the circle, $x^2+(y+6{)}^2=1$. Then the equation of the circle, passing through $C$ and having its centre at $P$ is:

Let O be the vertex and Q be any point on the parabola, ${\mathrm{x}}^2=8\mathrm{y}$. If the point P divides the line segment OQ internally in the ratio $1:3$, then the locus of P is :

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

Given: A circle, $2x^2+2y^2=5$ and a parabola, $y^2=4\sqrt{5}x$.

Statement-I : An equation of a common tangent to these curves is $\mathrm{y}=\mathrm{x}+\sqrt{5}$.

Statement - II: If the line, $y=mx+\frac{\sqrt{5}}{m}(m\ne 0)$ is their common tangent, then …

Statement 1: An equation of a common tangent to the parabola $y^2=16\sqrt{3}x$ and the ellipse $2x^2+y^2=4$ is $y=2x+2\sqrt{3}$.
Statement 2: If the line $y=mx+\frac{4\sqrt{3}}{m},(m\ne 0)$ is a common tangent to the parabola $y^2=16\sqrt{3}x$ and the …

If two tangents drawn from a point $P$ to the parabola $y^2=4x$ are at right angles, then the locus of $P$ is

A parabola has the origin as its focus and the line $\mathrm{x}=2$ as the directrix. Then the vertex of the parabola is at

The equation of a tangent to the parabola $y^2=8x$ is $y=x+2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

The locus of the vertices of the family of parabolas $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ is

Let $P$ be the point $(1,0)$ and $Q$ a point on the locus $y^2=8x$. The locus of mid point of $PQ$ is

A circle touches the $x$-axis and also touches the circle with centre at $(0,3)$ and radius 2. The locus of the centre of the circle is

A variable circle passes through the fixed point $A(p,q)$ and touches $x$-axis. The locus of the other end of the diameter through $A$ is

If $a\ne 0$ and the line $2bx+3cy+4d=0$ passes through the points of intersection of the parabolas $y^2=4ax$ and $x^2=4ay$, then