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Parabola

33 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Parabola

Parabola

33 questions · Mathematics · JEE Advanced

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 first…2025 · Shift 2 · Q22 · Multiple correct
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 the following…2024 · Shift 2 · Q24 · Multiple correct
A normal with slope $\frac{1}{6}$ is drawn from the point $(0, - α)$ to the parabola $x^{2} = - 4 a y$ , where $a > 0$ . Let $L$ be the line passing through $(0, - α)$ and parallel to the directrix of the parabola. Suppose that $L$ …2024 · Shift 2 · Q29 · Numerical
Let $P$ be a point on the parabola $y^{2} = 4 a x$ , where $a > 0$ . The normal to the parabola at $P$ meets the $x$ -axis at a point $Q$ . The area of the triangle $P F Q$ , where $F$ is the focus of the parabola, is 120 . If the slope $m$ of the…2023 · Shift 1 · Q24 · MCQ
Consider the parabola $y^{2} = 4 x$ . 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 $S P_{1}$ …2022 · Shift 1 · Q31 · Multiple correct
Let E denote the parabola y2 = 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 following statements is(are) TRUE?2021 · Shift 2 · Q25 · Multiple correct
Let a, b and $λ$ be positive real numbers. Suppose P is an end point of the latus return of the parabola y2 = 4 $λ$ x, and suppose the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ passes through the point…2020 · Shift 1 · Q22 · MCQ
Let the circles C1 : x2 + y2 = 9 and C2 : (x $-$ 3)2 + (y $-$ 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x $-$ h)2 + (y $-$ k)2 = r2 satisfies the following conditions : (i) Centre of C3 is collinear with… Includes diagram2019 · Shift 2 · Q35 · MCQ
Let the circle C1 : x2 + y2 = 9 and C2 : (x $-$ 3)2 + (y $-$ 4)2 = 16, intersect at the points X and Y. Suppose that another circle C3 : (x $-$ h)2 + (y $-$ k)2 = r2 satisfies the following conditions : (i) centre of C3 is collinear with… Includes diagram2019 · Shift 2 · Q36 · MCQ
If a chord, which is not a tangent, of the parabola y2 = 16x has the equation 2x + y = p, and mid-point (h, k), then which of the following is(are) possible value(s) of p, h and k?2017 · Shift 1 · Q25 · Multiple correct
By appropriately matching the information given in the three columns of the following table. Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. If a tangent to a suitable conic… Includes table2017 · Shift 1 · Q33 · MCQ
The circle $C_{1} : x^{2} + y^{2} = 3,$ with centre at $O$ , intersects the parabola $x^{2} = 2 y$ at the point $P$ in the first quadrant, Let the tangent to the circle $C_{1}$ , at $P$ touches other two circles $C_{2}$ and $C_{3}$ at $R_{2}$ …2016 · Shift 1 · Q27 · Multiple correct
Let $P$ be the point on the parabola $y^{2} = 4 x$ which is at the shortest distance from the center $S$ of the circle $x^{2} + y^{2} - 4 x - 16 y + 64 = 0$ . Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then2016 · Shift 2 · Q31 · Multiple correct
Let $P$ and $Q$ be distinct points on the parabola $y^{2} = 2 x$ such that a circle with $P Q$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $Δ O P Q$ is $32,$ …2015 · Shift 1 · Q24 · Multiple correct
If the normals of the parabola $y^{2} = 4 x$ 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 $r^{2}$ is2015 · Shift 1 · Q25 · Numerical
Let the curve $C$ be the mirror image of the parabola $y^{2} = 4 x$ 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 $B$ is2015 · Shift 1 · Q26 · Numerical
Suppose that the foci of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{5} = 1$ are $(f_{1}, 0)$ and $(f_{2}, 0)$ where $f_{1} > 0$ and $f_{2} < 0$ . Let $P_{1}$ and $P_{2}$ be two parabolas with a…2015 · Shift 2 · Q31 · Numerical
Let $a, r, s, t$ be nonzero real numbers. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ and $S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . Suppose that $P Q$ is…2014 · Shift 2 · Q26 · MCQ
Let $a, r, s, t$ be nonzero real numbers. Let $P (a t^{2}, 2 a t), Q, R (a r^{2}, 2 a r)$ and $S (a s^{2}, 2 a s)$ be distinct points on the parabola $y^{2} = 4 a x$ . Suppose that $P Q$ is…2014 · Shift 2 · Q32 · MCQ
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 $y = 2 x + a$ , $a > 0$ . If chord $P Q$ subtends an angle $θ$ at the vertex of $y^{2} = 4 a x$ , then tan $θ =$ …2013 · Shift 2 · Q31 · MCQ
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 $y = 2 x + a$ , $a > 0$ . Length of chord $P Q$ is2013 · Shift 2 · Q32 · MCQ
A line $L : y = m x + 3$ meets $y$ -axis at R $(0, 3)$ and the arc of the parabola $y^{2} = 16 x, 0 \leq y \leq 6$ at the point $F (x_{0}, y_{0})$ . The tangent to the parabola at $F (x_{0}, y_{0})$ intersects the $y$ -axis…2013 · Shift 2 · Q33 · MCQ
Let $S$ be the focus of the parabola $y^{2} = 8 x$ and let $P Q$ be the common chord of the circle $x^{2} + y^{2} - 2 x - 4 y = 0$ and the given parabola. The area of the triangle $P QS$ is2012 · Shift 1 · Q29 · Numerical
Consider the parabola $y^{2} = 8 x$ . Let $Δ_{1}$ be the area of the triangle formed by the end points of its latus rectum and the point $P (\frac{1}{2}, 2)$ on the parabola and $Δ_{2}$ be the area of the triangle…2011 · Shift 1 · Q32 · Numerical
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 $P$ is2011 · Shift 2 · Q26 · MCQ
Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by2011 · Shift 2 · Q38 · Multiple correct
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 be2010 · Shift 1 · Q35 · Multiple correct
The tangent $P T$ and the normal $P N$ to the parabola $y^{2} = 4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $P T N$ is a parabola whose2009 · Shift 2 · Q35 · Multiple correct
The locus of the orthocentre of the triangle formed by the lines $(1 + p) x - p y + p (1 + p) = 0, (1 + q) x - q y + q (1 + q) = 0$ and $y = 0$ , where $p e q$ , is :2009 · Shift 2 · Q38 · MCQ
The tangent to the curve $y = e^{x}$ drawn at the point ( $c, e^{c}$ ) intersects the line joining the points ( $c - 1, e^{c - 1}$ ) and ( $c + 1, e^{c + 1}$ )2007 · Shift 1 · Q26 · MCQ
Consider the circle $x^{2} + y^{2} = 9$ and the parabola $y^{2} = 8 x$ . They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola…2007 · Shift 1 · Q39 · MCQ
Consider the circle $x^{2} + y^{2} = 9$ and the parabola $y^{2} = 8 x$ . They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola…2007 · Shift 1 · Q40 · MCQ
Consider the circle $x^{2} + y^{2} = 9$ and the parabola $y^{2} = 8 x$ . They intersect at P and Q in the first and the fourth quadrants, respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents to the parabola…2007 · Shift 1 · Q41 · MCQ