New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Ellipse

66 questions · Mathematics · JEE Main

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Ellipse

Ellipse

66 questions · Mathematics · JEE Main

If $S$ and $S^{'}$ are the foci of the ellipse $\frac{x ^{2}}{18} + \frac{y ^{2}}{9} = 1$ and P be a point on the ellipse, then $min (S P \cdot S^{'} P) + max (S P \cdot S^{'} P)$ is equal to :2025 · 2 Apr · Shift 1 · Q30 · MCQ
If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :2025 · 2 Apr · Shift 2 · Q41 · MCQ
A line passing through the point $P (5, 5)$ intersects the ellipse $\frac{x ^{2}}{36} + \frac{y ^{2}}{25} = 1$ at $A$ and $B$ such that $(P A) \cdot (P B)$ is maximum. Then $5 (P A^{2} + P B^{2})$ is equal to :2025 · 3 Apr · Shift 1 · Q37 · MCQ
Let $C$ be the circle of minimum area enclosing the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ with eccentricity $\frac{1}{2}$ and foci $(\pm 2, 0)$ . Let $P QR$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the…2025 · 3 Apr · Shift 2 · Q30 · MCQ
The length of the latus-rectum of the ellipse, whose foci are $(2, 5)$ and $(2, - 3)$ and eccentricity is $\frac{4}{5}$ , is2025 · 4 Apr · Shift 1 · Q29 · MCQ
Let for two distinct values of p the lines $y = x + p$ touch the ellipse $E : \frac{x ^{2}}{4 ^{2}} + \frac{y ^{2}}{3 ^{2}} = 1$ at the points A and B . Let the line $y = x$ intersect E at the points C and D . Then the area of the quadrilateral…2025 · 4 Apr · Shift 2 · Q34 · MCQ
The centre of a circle C is at the centre of the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ . Let C pass through the foci $F_{1}$ and $F_{2}$ of E such that the circle $C$ and the…2025 · 4 Apr · Shift 2 · Q35 · MCQ
Let the length of a latus rectum of an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ be 10. If its eccentricity is the minimum value of the function $f (t) = t^{2} + t + \frac{11}{12}$ , $t \in R$ , then $a^{2} + b^{2}$ is equal to :2025 · 7 Apr · Shift 2 · Q33 · MCQ
Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If p + q = 126, then the…2025 · 7 Apr · Shift 2 · Q42 · MCQ
Let the ellipse $3 x^{2} + p y^{2} = 4$ pass through the centre $C$ of the circle $x^{2} + y^{2} - 2 x - 4 y - 11 = 0$ of radius $r$ . Let $f_{1}, f_{2}$ be the focal distances of the point $C$ on the ellipse. Then $6 f_{1} f_{2} - r$ is equal to2025 · 8 Apr · Shift 2 · Q32 · MCQ
Let $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ and $H : \frac{x ^{2}}{A ^{2}} - \frac{y ^{2}}{B ^{2}} = 1$ . Let the distance between the foci of E and the foci of $H$ be $23$ …2025 · 22 Jan · Shift 2 · Q45 · MCQ
The length of the chord of the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ , whose mid-point is $(1, \frac{1}{2})$ , is :2025 · 23 Jan · Shift 2 · Q28 · MCQ
Let the product of the focal distances of the point $(3, \frac{1}{2})$ on the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, (a > b)$ , be $\frac{7}{4}$ . Then the absolute difference of the eccentricities of two such ellipses…2025 · 24 Jan · Shift 1 · Q33 · MCQ
The equation of the chord, of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ , whose mid-point is $(3, 1)$ is :2025 · 24 Jan · Shift 2 · Q39 · MCQ
Let $E_{1} : \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ be an ellipse. Ellipses $E_{i}$ 's are constructed such that their centres and eccentricities are same as that of $E_{1}$ , and the length of minor axis of $E_{i}$ …2025 · 28 Jan · Shift 1 · Q49 · Numerical
If the midpoint of a chord of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ is $(2, 4/3)$ , and the length of the chord is $\frac{2 α}{3}$ , then $α$ is :2025 · 28 Jan · Shift 2 · Q29 · MCQ
Let the ellipse $E_{1} : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $a > b$ and $E_{2} : \frac{x ^{2}}{A ^{2}} + \frac{y ^{2}}{B ^{2}} = 1$ , $A < B$ have same eccentricity $\frac{1}{3}$ . Let the product of their lengths of latus rectums be $\frac{32}{3}$ …2025 · 29 Jan · Shift 1 · Q30 · MCQ
If $α x + β y = 109$ is the equation of the chord of the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ , whose mid point is $(\frac{5}{2}, \frac{1}{2})$ . then $α + β$ is equal to :2025 · 29 Jan · Shift 2 · Q36 · MCQ
Let $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ be an ellipse, whose eccentricity is $\frac{1}{2}$ and the length of the latusrectum is $14$ . Then the square of the eccentricity of $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ …2024 · 1 Feb · Shift 1 · Q46 · MCQ
Let $P$ be a point on the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ . Let the line passing through $P$ and parallel to $y$ -axis meet the circle $x^{2} + y^{2} = 9$ at point $Q$ such that $P$ and $Q$ are on…2024 · 1 Feb · Shift 2 · Q39 · MCQ
Let the line $2 x + 3 y - k = 0, k > 0$ , intersect the $x$ -axis and $y$ -axis at the points $A$ and $B$ , respectively. If the equation of the circle having the line segment $A B$ as a diameter is $x^{2} + y^{2} - 3 x - 2 y = 0$ …2024 · 5 Apr · Shift 1 · Q45 · MCQ
Let $f (x) = x^{2} + 9, g (x) = \frac{x}{x - 9}$ and $a = f \circ g (10), b = g \circ f (3)$ . If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x ^{2}}{a} + \frac{y ^{2}}{b} = 1$ …2024 · 9 Apr · Shift 1 · Q43 · MCQ
The length of the chord of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ , whose mid point is $(1, \frac{2}{5})$ , is equal to :2024 · 27 Jan · Shift 1 · Q38 · MCQ
If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :2024 · 30 Jan · Shift 1 · Q41 · MCQ
Let $A (α, 0)$ and $B (0, β)$ be the points on the line $5 x + 7 y = 50$ . Let the point $P$ divide the line segment $A B$ internally in the ratio $7 : 3$ . Let $3 x - 25 = 0$ be a directrix of the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ …2024 · 30 Jan · Shift 2 · Q50 · MCQ
Let $P$ be a parabola with vertex $(2, 3)$ and directrix $2 x + y = 6$ . Let an ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ , of eccentricity $\frac{1}{2}$ pass through the focus of the parabola $P$ . Then, the square of the…2024 · 31 Jan · Shift 2 · Q36 · MCQ
In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $α$ and the number of persons who speak only Hindi is $β$ ,…2023 · 6 Apr · Shift 2 · Q33 · MCQ
Let the ellipse $E : x^{2} + 9 y^{2} = 9$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If…2023 · 10 Apr · Shift 1 · Q26 · MCQ
Consider ellipses $E_{k} : k x^{2} + k^{2} y^{2} = 1, k = 1, 2, \dots, 20$ . Let $C_{k}$ be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $E_{k}$ …2023 · 11 Apr · Shift 1 · Q35 · MCQ
Let $P (\frac{2 3}{7}, \frac{6}{7}), Q, R$ and $S$ be four points on the ellipse $9 x^{2} + 4 y^{2} = 36$ . Let $PQ$ and $RS$ be mutually perpendicular and…2023 · 12 Apr · Shift 1 · Q31 · MCQ
Let an ellipse with centre $(1, 0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $x$ -axis. If its minor axis subtends an angle $6 0^{\circ}$ at the foci, then the square of the sum of the lengths of its minor…2023 · 15 Apr · Shift 1 · Q39 · Numerical
Let C be the largest circle centred at (2, 0) and inscribed in the ellipse $\frac{x ^{2}}{36} + \frac{y ^{2}}{16} = 1$ . If (1, $α$ ) lies on C, then 10 $α^{2}$ is equal to 2023 · 24 Jan · Shift 1 · Q40 · Numerical
Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{4} = 1, a > 2$ , having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel…2022 · 24 Jun · Shift 2 · Q29 · MCQ
If the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ meets the line $\frac{x}{7} + \frac{y}{2 6} = 1$ on the $x$ -axis and the line $\frac{x}{7} - \frac{y}{2 6} = 1$ on the $y$ -axis, then the eccentricity of the ellipse is :2022 · 25 Jul · Shift 2 · Q31 · MCQ
The line y = x + 1 meets the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ at two points P and Q. If r is the radius of the circle with PQ as diameter then (3r)2 is equal to :2022 · 25 Jun · Shift 2 · Q36 · MCQ
The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse $x^{2} + 2 y^{2} = 4$ is an ellipse with eccentricity :2022 · 26 Jun · Shift 2 · Q34 · MCQ
If the length of the latus rectum of the ellipse $x^{2} + 4 y^{2} + 2 x + 8 y - λ = 0$ is 4 , and $l$ is the length of its major axis, then $λ + l$ is equal to .2022 · 27 Jul · Shift 1 · Q45 · Numerical
Let the eccentricity of an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $a > b$ , be $\frac{1}{4}$ . If this ellipse passes through the point $(- 4 \frac{2}{5}, 3)$ , then $a^{2} + b^{2}$ is…2022 · 27 Jun · Shift 1 · Q30 · MCQ
Let a line L pass through the point of intersection of the lines $b x + 10 y - 8 = 0$ and $2 x - 3 y = 0, b \in R - {\frac{4}{3}}$ . If the line $L$ also passes through the point $(1, 1)$ and touches the circle…2022 · 29 Jul · Shift 1 · Q32 · MCQ
Let the eccentricity of the ellipse $x^{2} + a^{2} y^{2} = 25 a^{2}$ be b times the eccentricity of the hyperbola $x^{2} - a^{2} y^{2} = 5$ , where a is the minimum distance between the curves y = ex and y = logex. Then $a^{2} + \frac{1}{b ^{2}}$ …2022 · 30 Jun · Shift 1 · Q32 · MCQ
Let $E_{1} : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ . Let E2 be another ellipse such that it touches the end points of major axis of E1 and the foci of E2 are the end points of minor axis of E1. If E1 and E2 have…2021 · 22 Jul · Shift 2 · Q37 · MCQ
If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is :2021 · 25 Feb · Shift 2 · Q38 · MCQ
Let an ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $a^{2} > b^{2}$ , passes through $(\frac{3}{2}, 1)$ and has eccentricity $\frac{1}{3}$ . If a circle, centered at focus…2021 · 25 Jul · Shift 1 · Q37 · MCQ
If x2 + 9y2 $-$ 4x + 3 = 0, x, y $\in$ R, then x and y respectively lie in the intervals :2021 · 27 Aug · Shift 1 · Q31 · MCQ
A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from…2021 · 27 Jul · Shift 1 · Q28 · MCQ
The locus of mid-points of the line segments joining ( $-$ 3, $-$ 5) and the points on the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1$ is :2021 · 31 Aug · Shift 2 · Q27 · MCQ
Let $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ (a > b) be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, $$\phi \left( t \right) = {5 \over {12}} + t -…2020 · 4 Sep · Shift 1 · Q37 · MCQ
If the co-ordinates of two points A and B are $(7, 0)$ and $(- 7, 0)$ respectively and P is any point on the conic, 9x2 + 16y2 = 144, then PA + PB is equal to :2020 · 5 Sep · Shift 1 · Q26 · MCQ
If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is :2020 · 7 Jan · Shift 1 · Q41 · MCQ
In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0,5 $3$ ), then the length of its latus rectum is :2019 · 8 Apr · Shift 2 · Q26 · MCQ
Let S = ${(x, y) \in R^{2} : \frac{y ^{2}}{1 + r} - \frac{x ^{2}}{1 - r}}; r e \pm 1.$ Then S represents :2019 · 10 Jan · Shift 2 · Q33 · MCQ
Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following…2019 · 11 Jan · Shift 2 · Q24 · MCQ
An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the following points?2019 · 12 Apr · Shift 2 · Q37 · MCQ
Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $Δ$ S'BS is a right angled triangle with right angle at B and area ( $Δ$ S'BS) = 8 sq. units, then the length of a latus rectum of…2019 · 12 Jan · Shift 2 · Q43 · MCQ
If the length of the latus rectum of an ellipse is 4 units and the distance between a focus an its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is :2018 · 16 Apr · Shift 1 · Q45 · MCQ
Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is $\frac{3}{5}$ and the distance between its foci is 6, then the area (in sq. units) of the quadrilatateral inscribed in the…2017 · 8 Apr · Shift 1 · Q34 · MCQ
The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points (4, −1) and (−2, 2) is :2017 · 9 Apr · Shift 1 · Q37 · MCQ
The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2} + 3 y^{2} = 6$ on any tangent to it is :2014 · Shift 0 · Q36 · MCQ
The equation of the circle passing through the foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ , and having centre at $(0, 3)$ is :2013 · Shift 0 · Q36 · MCQ
An ellipse is drawn by taking a diameter of thec circle $(x - 1)^{2} + y^{2} = 1$ as its semi-minor axis and a diameter of the circle $x^{2} + (y - 2)^{2} = 4$ is semi-major axis. If the centre of the…2012 · Shift 0 · Q37 · MCQ
Equation of the ellipse whose axes of coordinates and which passes through the point $(- 3, 1)$ and has eccentricity $\frac{2}{5}$ is :2011 · Shift 0 · Q44 · MCQ
The ellipse $x^{2} + 4 y^{2} = 4$ is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscribed in another ellipse that passes through the point $(4, 0)$ . Then the equation of the ellipse is :2009 · Shift 0 · Q38 · MCQ
A focus of an ellipse is at the origin. The directrix is the line $x = 4$ and the eccentricity is $\frac{1}{2}$ . Then the length of the semi-major axis is :2008 · Shift 0 · Q45 · MCQ
In the ellipse, the distance between its foci is $6$ and minor axis is $8$ . Then its eccentricity is :2006 · Shift 0 · Q81 · MCQ
An ellipse has $O B$ as semi minor axis, $F$ and $F$ ' its focii and theangle $F B F$ ' is a right angle. Then the eccentricity of the ellipse is :2005 · Shift 0 · Q107 · MCQ
The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$ . If one of the directrices is $x = 4$ , then the equation of the ellipse is :2004 · Shift 0 · Q84 · MCQ