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Parabola

69 questions · Mathematics · JEE Main

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Parabola

Parabola

69 questions · Mathematics · JEE Main

Let the focal chord PQ of the parabola $y^{2} = 4 x$ make an angle of $6 0^{\circ}$ with the positive $x$ axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the $y$ -axis…2025 · 2 Apr · Shift 1 · Q31 · MCQ
Let the point P of the focal chord PQ of the parabola $y^{2} = 16 x$ be $(1, - 4)$ . If the focus of the parabola divides the chord $P Q$ in the ratio $m : n, gcd (m, n) = 1$ , then $m^{2} + n^{2}$ is equal to :2025 · 2 Apr · Shift 2 · Q29 · MCQ
The radius of the smallest circle which touches the parabolas $y = x^{2} + 2$ and $x = y^{2} + 2$ is2025 · 3 Apr · Shift 1 · Q31 · MCQ
A line passing through the point $A (- 2, 0)$ , touches the parabola $P : y^{2} = x - 2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $A B$ , parabola $P$ and the $x$ -axis, is :2025 · 4 Apr · Shift 2 · Q36 · MCQ
The axis of a parabola is the line $y = x$ and its vertex and focus are in the first quadrant at distances $2$ and $22$ units from the origin, respectively. If the point $(1, k)$ lies on the parabola, then a possible value of…2025 · 4 Apr · Shift 2 · Q41 · MCQ
Let P be the parabola, whose focus is $(- 2, 1)$ and directrix is $2 x + y + 2 = 0$ . Then the sum of the ordinates of the points on P, whose abscissa is $-$ 2, is2025 · 7 Apr · Shift 1 · Q28 · MCQ
Let $r$ be the radius of the circle, which touches $x$ - axis at point $(a, 0), a < 0$ and the parabola $y^{2} = 9 x$ at the point $(4, 6)$ . Then $r$ is equal to .2025 · 8 Apr · Shift 2 · Q48 · Numerical
Let the parabola $y = x^{2} + p x - 3$ , meet the coordinate axes at the points $(- 1, - 1)$ and R . If the circle C with centre at $(- 1, - 1)$ passes through the points $R$ and $R$ , then the area of $△ P QR$ is :2025 · 22 Jan · Shift 1 · Q41 · MCQ
Let $P (4, 43)$ be a point on the parabola $y^{2} = 4 a x$ and PQ be a focal chord of the parabola. If M and N are the foot of perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the…2025 · 22 Jan · Shift 2 · Q44 · MCQ
If the line $3 x - 2 y + 12 = 0$ intersects the parabola $4 y = 3 x^{2}$ at the points $A$ and $B$ , then at the vertex of the parabola, the line segment AB subtends an angle equal to2025 · 23 Jan · Shift 1 · Q40 · MCQ
Let the shortest distance from $(a, 0), a > 0$ , to the parabola $y^{2} = 4 x$ be 4 . Then the equation of the circle passing through the point $(a, 0)$ and the focus of the parabola, and having its centre on the axis of the parabola is :2025 · 23 Jan · Shift 2 · Q44 · MCQ
The focus of the parabola $y^{2} = 4 x + 16$ is the centre of the circle $C$ of radius 5 . If the values of $λ$ , for which C passes through the point of intersection of the lines $3 x - y = 0$ and $x + λ y = 4$ , are $λ_{1}$ and $λ_{2}, λ_{1} < λ_{2}$ …2025 · 23 Jan · Shift 2 · Q47 · Numerical
If the equation of the parabola with vertex $V (\frac{3}{2}, 3)$ and the directrix $x + 2 y = 0$ is $α x^{2} + β y^{2} - γ x y - 30 x - 60 y + 225 = 0$ , then $α + β + γ$ is equal to :2025 · 24 Jan · Shift 2 · Q37 · MCQ
Let ABCD be a trapezium whose vertices lie on the parabola $y^{2} = 4 x$ . Let the sides AD and BC of the trapezium be parallel to $y$ -axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point…2025 · 28 Jan · Shift 1 · Q37 · MCQ
Let $A$ and $B$ be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y^{2} = 4 x$ with respect to the line $x + y + 4 = 0$ . If $d$ denotes the distance between $A$ and $B$ , and a denotes the area of $△ S A B$ …2025 · 28 Jan · Shift 2 · Q47 · Numerical
Two parabolas have the same focus (4, 3) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at the points A and B, then (AB)2 is equal to :2025 · 29 Jan · Shift 1 · Q31 · MCQ
Let $y^{2} = 12 x$ be the parabola and $S$ be its focus. Let $P Q$ be a focal chord of the parabola such that $(S P) (S Q) = \frac{147}{4}$ . Let $C$ be the circle described taking $P Q$ as a diameter. If the equation of a circle $C$ is $64 x^{2} + 64 y^{2} - α x - 643 y = β$ …2025 · 29 Jan · Shift 2 · Q47 · Numerical
Let the line $L : 2 x + y = α$ pass through the point of the intersection $P$ (in the first quadrant) of the circle $x^{2} + y^{2} = 3$ and the parabola $x^{2} = 2 y$ . Let the line $L$ touch two circles $C_{1}$ …2024 · 1 Feb · Shift 1 · Q56 · Numerical
Let the length of the focal chord PQ of the parabola $y^{2} = 12 x$ be 15 units. If the distance of $PQ$ from the origin is $p$ , then $10 p^{2}$ is equal to .2024 · 4 Apr · Shift 1 · Q58 · Numerical
Let $P Q$ be a chord of the parabola $y^{2} = 12 x$ and the midpoint of $P Q$ be at $(4, 1)$ . Then, which of the following point lies on the line passing through the points $P$ and $Q$ ?2024 · 4 Apr · Shift 2 · Q43 · MCQ
Suppose $AB$ is a focal chord of the parabola $y^{2} = 12 x$ of length $l$ and slope $m < 3$ . If the distance of the chord $AB$ from the origin is $d$ , then $l d^{2}$ is equal to …2024 · 5 Apr · Shift 1 · Q53 · Numerical
Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y^{2} = 4 (x - 9)$ at the point P. The distance of the point P from the centre of the circle $x^{2} + y^{2} - 14 x - 8 y + 56 = 0$ is .2024 · 5 Apr · Shift 2 · Q60 · Numerical
Let $C$ be the circle of minimum area touching the parabola $y = 6 - x^{2}$ and the lines $y = 3 ∣ x ∣$ . Then, which one of the following points lies on the circle $C$ ?2024 · 6 Apr · Shift 1 · Q34 · MCQ
Let a conic $C$ pass through the point $(4, - 2)$ and $P (x, y), x \geq 3$ , be any point on $C$ . Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3, - 5)$ . If…2024 · 6 Apr · Shift 1 · Q52 · Numerical
Let $L_{1}, L_{2}$ be the lines passing through the point $P (0, 1)$ and touching the parabola $9 x^{2} + 12 x + 18 y - 14 = 0$ . Let $Q$ and $R$ be the points on the lines $L_{1}$ and $L_{2}$ such that the $△ P QR$ is an isosceles triangle with base $QR$ …2024 · 6 Apr · Shift 1 · Q60 · Numerical
Let $C$ and $C$ be three points on the parabola $y^{2} = 6 x$ and let the line segment $A B$ meet the line $L$ through $C$ parallel to the $x$ -axis at the point $D$ . Let $M$ and $N$ respectively be the feet of the perpendiculars from $A$ …2024 · 9 Apr · Shift 2 · Q52 · Numerical
Consider the circle $C : x^{2} + y^{2} = 4$ and the parabola $P : y^{2} = 8 x$ . If the set of all values of $α$ , for which three chords of the circle $C$ on three distinct lines passing through the point $(α, 0)$ are bisected by the parabola $P$ …2024 · 9 Apr · Shift 2 · Q55 · Numerical
If the shortest distance of the parabola $y^{2} = 4 x$ from the centre of the circle $x^{2} + y^{2} - 4 x - 16 y + 64 = 0$ is $d$ , then $d^{2}$ is equal to :2024 · 27 Jan · Shift 1 · Q34 · MCQ
Let $P (α, β)$ be a point on the parabola $y^{2} = 4 x$ . If $P$ also lies on the chord of the parabola $x^{2} = 8 y$ whose mid point is $(1, \frac{5}{4})$ , then $(α - 28) (β - 8)$ is equal to .2024 · 29 Jan · Shift 2 · Q57 · Numerical
If the $x$ -intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3, then the length of this chord is equal to .2023 · 1 Feb · Shift 2 · Q40 · Numerical
Let $R$ be the focus of the parabola $y^{2} = 20 x$ and the line $y = m x + c$ intersect the parabola at two points $P$ and $Q$ . Let the point $G (10, 10)$ be the centroid of the triangle $P QR$ . If $c - m = 6$ , then $(P Q)^{2}$ is :2023 · 8 Apr · Shift 1 · Q29 · MCQ
Let $PQ$ be a focal chord of the parabola $y^{2} = 36 x$ of length 100 , making an acute angle with the positive $x$ -axis. Let the ordinate of $P$ be positive and $M$ be the point on the line segment PQ such that…2023 · 13 Apr · Shift 1 · Q24 · MCQ
The equations of two sides of a variable triangle are $x = 0$ and $y = 3$ , and its third side is a tangent to the parabola $y^{2} = 6 x$ . The locus of its circumcentre is :2023 · 25 Jan · Shift 2 · Q22 · MCQ
The parabolas : $a x^{2} + 2 b x + cy = 0$ and $d x^{2} + 2 e x + f y = 0$ intersect on the line $y = 1$ . If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then :2023 · 30 Jan · Shift 2 · Q31 · MCQ
A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :2022 · 24 Jun · Shift 2 · Q28 · MCQ
Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = .2022 · 24 Jun · Shift 2 · Q45 · Numerical
Let $x = 2 t$ , $y = \frac{t ^{2}}{3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $S A ⊥ B A$ , where A is any point on the conic. If k is the ordinate of the centroid of the $Δ$ SAB, then $t \to 1 lim k$ …2022 · 25 Jun · Shift 1 · Q37 · MCQ
Let $P$ and $Q$ be any points on the curves $(x - 1)^{2} + (y + 1)^{2} = 1$ and $y = x^{2}$ , respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of $P$ in the interval :2022 · 26 Jul · Shift 2 · Q23 · MCQ
If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $3 x + y - 29 = 0$ , is $x^{2} + a y^{2} + b x y + c x + d y + k = 0$ , then $a + b + c + d + k$ is equal to :2022 · 27 Jun · Shift 2 · Q29 · MCQ
If vertex of a parabola is (2, $-$ 1) and the equation of its directrix is 4x $-$ 3y = 21, then the length of its latus rectum is :2022 · 28 Jun · Shift 2 · Q36 · MCQ
Let the focal chord of the parabola $P : y^{2} = 4 x$ along the line $L : y = m x + c, m > 0$ meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $H : x^{2} - y^{2} = 4$ …2022 · 29 Jul · Shift 1 · Q37 · MCQ
Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of $\frac{π}{2}$ at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} =…2022 · 29 Jun · Shift 1 · Q34 · MCQ
Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $Δ$ POQ is equal to .2022 · 30 Jun · Shift 1 · Q39 · Numerical
Let P be a variable point on the parabola $y = 4 x^{2} + 1$ . Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is :2021 · 20 Jul · Shift 2 · Q36 · MCQ
If the point on the curve y2 = 6x, nearest to the point $(3, \frac{3}{2})$ is ( $α$ , $β$ ), then 2( $α$ + $β$ ) is equal to .2021 · 20 Jul · Shift 2 · Q43 · Numerical
The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a moving point of the parabola, is another parabola whose directrix is :2021 · 24 Feb · Shift 1 · Q34 · MCQ
If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x $-$ 1, then the co-ordinates of P are :2021 · 24 Feb · Shift 2 · Q30 · MCQ
The shortest distance between the line x $-$ y = 1 and the curve x2 = 2y is :2021 · 25 Feb · Shift 2 · Q24 · MCQ
If two tangents drawn from a point P to the parabola y2 = 16(x $-$ 3) are at right angles, then the locus of point P is :2021 · 27 Aug · Shift 2 · Q26 · MCQ
The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :2021 · 31 Aug · Shift 1 · Q32 · MCQ
The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the vertex of this parabola, is :2020 · 2 Sep · Shift 2 · Q28 · MCQ
Let P be a point on the parabola, y2 = 12x and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the…2020 · 3 Sep · Shift 1 · Q31 · MCQ
Let the latus ractum of the parabola y2 = 4x be the common chord to the circles C1 and C2 each of them having radius 2 $5$ . Then, the distance between the centres of the circles C1 and C2 is :2020 · 3 Sep · Shift 2 · Q24 · MCQ
The locus of a point which divides the line segment joining the point (0, –1) and a point on the parabola, x2 = 4y, internally in the ratio 1 : 2, is :2020 · 8 Jan · Shift 1 · Q29 · MCQ
If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is :2019 · 9 Apr · Shift 1 · Q32 · MCQ
Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it?2019 · 9 Jan · Shift 1 · Q33 · MCQ
If $θ$ denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, the |tan $θ$ | is equal to :2019 · 9 Jan · Shift 1 · Q44 · MCQ
Let A(4, $-$ 4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of $Δ$ ACB is maximum. Then, the area (in sq. units) of $Δ$ ACB, is :2019 · 9 Jan · Shift 2 · Q26 · MCQ
The length of the chord of the parabola x2 $=$ 4y having equation x – $2 y + 42 = 0$ is -2019 · 10 Jan · Shift 2 · Q45 · MCQ
If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x – a2) = 0 and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is :2019 · 11 Jan · Shift 2 · Q39 · MCQ
The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is :2019 · 12 Jan · Shift 1 · Q29 · MCQ
Let P(4, –4) and Q(9, 6) be two points on the parabola, y2 = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $Δ$ PXQ is maximum. Then this maximum area (in sq.…2019 · 12 Jan · Shift 1 · Q31 · MCQ
Tangents drawn from the point ( $-$ 8, 0) to the parabola y2 = 8x touch the parabola at $P$ and $Q .$ If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to :2018 · 15 Apr · Shift 2 · Q34 · MCQ
Let $O$ be the vertex and $Q$ be any point on the parabola, $x^{2} = 8 y$ . If the point $P$ divides the line segment $O Q$ internally in the ratio $1 : 3$ , then locus of $P$ is :2015 · Shift 0 · Q36 · MCQ
If two tangents drawn from a point $P$ to the parabola $y^{2} = 4 x$ are at right angles, then the locus of $P$ is2010 · Shift 0 · Q41 · MCQ
A parabola has the origin as its focus and the line $x = 2$ as the directrix. Then the vertex of the parabola is at :2008 · Shift 0 · Q44 · MCQ
The locus of the vertices of the family of parabolas $y = \frac{a ^{3} x ^{2}}{3} + \frac{a ^{2} x}{2} - 2 a$ is :2006 · Shift 0 · Q80 · MCQ
Let $P$ be the point $(1, 0)$ and $Q$ a point on the parabola $y^{2} = 8 x$ . The locus of mid point of $P Q$ is :2005 · Shift 0 · Q108 · MCQ
If $a e 0$ and the line $2 b x + 3 cy + 4 d = 0$ passes through the points of intersection of the parabolas $y^{2} = 4 a x$ and $x^{2} = 4 a y$ , then :2004 · Shift 0 · Q112 · MCQ