Parabola — JEE Advanced PYQs

Let T be the tangent to the parabola ${\mathrm{y}}^2=16\mathrm{x}$ at the point (64, 32). Let L be the tangent to the same parabola at another point $\left({\mathrm{x}}_1,{\mathrm{y}}_1\right)$ on the parabola. If L and T are perpendicular to each other, then the distance betwe …

Let $A_1,B_1,C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1{C}_1$ and $B_1{C}_1$ are tangents to the curve $y^2=8x$ at $A_1$ and $B_1$, respectively. If $O=(0,0)$ and $C_1=(-4,0)$, then which of the following statements is (are) TRUE?

A normal with slope $\frac{1}{\sqrt{6}}$ is drawn from the point $(0,-\alpha )$ to the parabola $x^2=-4ay$, where $a>0$. Let $L$ be the line passing through $(0,-\alpha )$ and parallel to the directrix of the parabola.
Suppose that $L$ intersects the parabola at two points $A$ …

Let ${\mathrm{T}}_1$ and ${\mathrm{T}}_2$ be two distinct common tangents to the ellipse $\mathrm{E}:\frac{{\mathrm{x}}^2}{6}+\frac{{\mathrm{y}}^2}{3}=1$ and the parabola $\mathrm{P}:{\mathrm{y}}^2=12\mathrm{x}$. Suppose that the tangent ${\mathrm{T}}_1$ touches P and E at the …

Let P be a point on the parabola ${\mathrm{y}}^2=4\mathrm{ax}$, where $\mathrm{a}>0$. 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 slope m of the normal and a are both positive i …

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 the lines $S{P}_1$ and $S{P}_2$ respectively such that …

Let E denote the parabola ${\mathrm{y}}^2=8\mathrm{x}$. Let $\mathrm{P}=(-2,4)$, 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 following statements is (are) TRUE?

If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint $(h,k)$, then which of the following is(are) possible value(s) of $p,h$ and $k$?

| (II) $x^2+a^2{y}^2=a^2$ | (ii) $y=mx+a\sqrt{m^2+1}$ | (Q) …

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

Suppose that the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ are $\left(f_1,0\right)$ and $\left(f_2,0\right)$ where $f_1>0$ and $f_2<0$. Let $P_1$ and $P_2$ be two parabolas with a common vertex at (0, 0) and with foci at …

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between $A$ and $B$ is

If the normals of the parabola $y^2=4x$ drawn at the end points of its latus rectum are tangents to the circle $(x-3{)}^2+(y+2{)}^2=r^2$, then the value of $r^2$ is

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 $△OPQ$ is $3\sqrt{2}$, then which of the following is …

Let $a,r,s,t$ be nonzero real numbers. Let $P\left(a{t}^2,2at\right),Q,R\left(a{r}^2,2ar\right)$ and $S\left(a{s}^2,2as\right)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are par …

Let $a,r,s,t$ be nonzero real numbers. Let $P\left(a{t}^2,2at\right),Q,R\left(a{r}^2,2ar\right)$ and $S\left(a{s}^2,2as\right)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are par …

Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,a>0$.

Length of chord $PQ$ is

Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,a>0$.

If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax$, then $\tan \theta =$

A line $L:y=mx+3$ meets $y$-axis at $E(0,3)$ and the arc of the parabola $y^2=16x$, $0\le y\le 6$ at the point $F\left(x_0,y_0\right)$. The tangent to the parabola at $F\left(x_0,y_0\right)$ intersects the $y$-axis at $G\left(0,y_1\right)$. The slope $m$ of …

Let $S$ be the focus of the parabola $y^2=8x$ and let $PQ$ be the common chord of the circle $x^2+y^2-2x-4y=0$ and the given parabola. The area of the triangle $PQS$ is

Consider the parabola $y^2=8x$. Let ${\Delta }_1$ be the area of the triangle formed by the end points of its latus rectum and the point $P\left(\frac{1}{2},2\right)$ on the parabola, and ${\Delta }_2$ be the area of the triangle formed by drawing tangents at $P$ and …

Let $(x,y)$ be any point on the parabola $y^2=4x$. 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 $L$ be a normal to the parabola $y^2=4x$. If $L$ passes through the point (9, 6), then $L$ is given by

The tangent $PT$ and the normal $PN$ to the parabola $y^2=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose

Consider the two curves

Then,

STATEMENT-1 : The curve $y=\frac{-x^2}{2}+x+1$ is symmetric with respect to the line $x=1$.

because

STATEMENT-2 : A parabola is symmetric about its axis.

The axis of a parabola is along the line $y=x$ and the distance of its vertex from origin is $\sqrt{2}$ and that from its focus is $2\sqrt{2}$. If vertex and focus both lie in the first quadrant, then the equation of the parabola is

The equations of the common tangents to the parabola $y=x^2$ and $y=-(x-2{)}^2$ is/are

Normals are drawn at points P, Q and R lying on the parabola $y^2=4x$ which intersect at (3, 0). Then
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