Parabola

Let $S$ denote the locus of the mid-points of those chords of the parabola $y^2=x$, such that the area of the region enclosed between the parabola and the chord is $\frac{4}{3}$. Let $R$ denote the region lying in the…

Let $A_1, B_1, C_1$ be three points in the $x y$-plane. Suppose that the lines $A_1 C_1$ and $B_1 C_1$ are tangents to the curve $y^2=8 x$ at $A_1$ and $B_1$, respectively. If $O=(0,0)$ and $C_1=(-4,0)$, then which of…

Let $P$ be a point on the parabola $y^{2}=4ax$, where $a>0$. The normal to the parabola at $P$ meets the $x$-axis at a point $Q$. The area of the triangle $PFQ$, where $F$ is the focus of the parabola, is $120$. If the…

Consider the parabola $y^{2}=4x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P=(-2,1)$ meet the parabola at $P_{1}$ and $P_{2}$. Let $Q_{1}$ and $Q_{2}$ be points on…

Let $E$ denote the parabola $y^{2}=8x.$ Let $P=(-2,4)$, and let $Q$ and $Q^{'}$ be two distinct points on $E$ such that the lines $PQ$ and $PQ^{'}$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the…

Let P be the point on the parabola $y^{2}=4x$ which is at the shortest distance from the center S of the circle $x^{2}+y^{2}-4x-16y+64=0.$ Let Q be the point on the circle dividing the line segment SP internally. Then -

Let $P$ and $Q$ be distinct points on the parabola $y^{2}=2x$ 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…

The common tangents to the circle $x^{2}+y^{2}=2$ and the parabola $y^{2}=8x$ touch the circle at the points P, and Q the parabola at the points R, S. Then the area of the quadrilateral PQRS is

Paragraph: Let $P Q$ be a focal chord of the parabola $y^{2}=4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2 x+a, a>0$. Question: If chord $P Q$ subtends an angle $\theta$ at…

Paragraph: Let $P Q$ be a focal chord of the parabola $y^{2}=4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2 x+a, a>0$. Question: Length of chord $P Q$ is

Let $(x, y)$ be any point on the parabola $y^2=4 x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x, y)$ in the ratio $1: 3$. Then, the locus of $P$ is

Let $A$ and $B$ be two distinct points on the parabola $y^2=4 x$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining $A$ and $B$ can be