Quantrex Academy Quantrex AcademyPYQs with solutions · JEE · NEET · NDA

Home · JEE PYQs · Mathematics · Ellipse

JEE Ellipse PYQs with Solutions

313 previous year questions from Ellipse (JEE Main & Advanced Mathematics) with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

313 questionsMathematics2003–2026Answers + solutions

About Ellipse PYQs

Ellipse has 313 previous year questions in the Quantrex Academy JEE Main & Advanced Mathematics bank. The 220 questions in this list come from JEE Main (129), NTA Abhyas JEE Main (53), JEE Advanced (22) and 1 other source, spanning 2003 to 2026. Each question opens on its own page with the options, the correct answer and the worked solution where available.

Questions by year (this list)

  • 2026: 16
  • 2025: 19
  • 2024: 9
  • 2023: 11
  • 2022: 12
  • 2021: 17
  • 2020: 63
  • 2019: 13
  • 2018: 5
  • 2017: 5
  • 2016: 2
  • 2015: 4

Ellipse questions with solutions

1

Consider the ellipse E given by (x^2)/(18) + (y^2)/(12) = 1 . Let H be the hyperbola whose eccentricity is the reciprocal of the eccentricity of E and whose foci are the same as

JEE Advanced 2026 Solution View → 2

Passage: Consider the ellipses given by x^2 + 4y^2 = 1 and 4x^2 + y^2 = 1 . Question: Let P be the point in the first quadrant where the given ellipses intersect. If θ is the

JEE Advanced 2026 Solution View → 3

Let (x^2)/(f(a^2+7a+3)) + (y^2)/(f(3a+15)) = 1 represent an ellipse with major axis along y -axis, where f is a strictly decreasing positive function on ℝ . If the set of all

JEE Main 2026 Solution View → 4

The eccentricity of an ellipse E with centre at the origin O is 3 2 and its directrices are x = ± 4 6 3 . Let H: (x^2)/(a^2) - (y^2)/(b^2) = 1 be a hyperbola whose eccentricity is

JEE Main 2026 Solution View → 5

Let x = 9 be a directrix of an ellipse E , whose centre is at the origin and eccentricity is (1)/(3) . Let P(α, 0) , α > 0 , be a focus of E and AB be a chord passing through P .

JEE Main 2026 Solution View → 6

Let a focus of the ellipse E: (x^2)/(a^2) + (y^2)/(b^2) = 1 be S(4, 0) and its eccentricity be (4)/(5) . If the point P(3, α) lies on E and O is the origin, then the area of POS

JEE Main 2026 Solution View → 7

Let A be the point (3, 0) and circles with variable diameter AB touch the circle x^2 + y^2 = 36 internally. Let the curve C be the locus of the point B. If the eccentricity of C

JEE Main 2026 Solution View → 8

Let an ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 , a < b , pass through the point (4, 3) and have eccentricity 5 3 . Then the length of its latus rectum is :

JEE Main 2026 Solution View → 9

An ellipse has its center at (1,-2) , one focus at (3,-2) and one vertex at (5,-2) . Then the length of its latus rectum is :

JEE Main 2026 Solution View → 10

Let the ellipse E: x^ 2 144 + y^ 2 169 =1 and the hyperbola H: x^ 2 16 - y^ 2 λ^ 2 =-1 have the same foci. If e and L respectively denote the eccentricity and the length of the

JEE Main 2026 Solution View → 11

Let the length of the latus rectum of an ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1,(a>b) , be 30. If its eccentricity is the maximum value of the function f(t)=-(3)/(4)+2 t-t^ 2 , then (

JEE Main 2026 Solution View → 12

Let (h, k) lie on the circle C: x^ 2 +y^ 2 =4 and the point (2 h+1,3 k+2) lie on an ellipse with eccentricity e . Then the value of 5 e^ 2 is equal to \_\_\_\_ .

JEE Main 2026 Solution View → 13

Let each of the two ellipses E_ 1 : x^ 2 a^ 2 + y^ 2 b^ 2 =1,(a>b) and E_ 2 : x^ 2 ~A^ 2 + y^ 2 ~B^ 2 =1,(~A<B) have eccentricity (4)/(5) . Let the lengths of the latus recta of

JEE Main 2026 Solution View → 14

Let the line y-x=1 intersect the ellipse x^ 2 2 + y^ 2 1 =1 at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is :

JEE Main 2026 Solution View → 15

Let S and S^ be the foci of the ellipse x^ 2 25 + y^ 2 9 =1 and P(α, β) be a point on the ellipse in the first quadrant. If (SP)^ 2 +(S^ P)^ 2 -SP · S^ P=37 , then α^ 2 +β^ 2 is

JEE Main 2026 Solution View → 16

If the line α x+4 y= 7 , where α ∈ R , touches the ellipse 3 x^ 2 +4 y^ 2 =1 at the point P in the first quadrant, then one of the focal distances of P is :

JEE Main 2026 Solution View → 17

Let P(x_1, y_1) and Q(x_2, y_2) be two distinct points on the ellipse (x^2)/(9)+(y^2)/(4)=1 such that y_1>0 , and y_2>0 . Let C denote the circle x^2+y^2=9 , and M be the point

JEE Advanced 2025 Solution View → 18

Let the ellipse 3 x^2+py^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

JEE Main 2025 Solution View → 19

LetA=\ (α, β) ∈ R × R :|α-1| ≤ 4 and |β-5| ≤ 6\ and B= \ (α, β) ∈ R × R : 16(α-2)^2+9(β-6)^2 ≤ 144\

JEE Main 2025 Solution View → 20

Let the length of a latus rectum of an ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 be 10 . If its eccentricity is the minimum value of the function f(t)=t^2+t+(11)/(12) , t ∈ R , then

JEE Main 2025 Solution View → 21

The centre of a circle C is at the centre of the ellipse E: (x^2)/(a^2)+(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 ellipse E

JEE Main 2025 Solution View → 22

Let for two distinct values of p the lines y=x+p touch the ellipse E: x^2 4^2 + 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

JEE Main 2025 Solution View → 23

The length of the latus-rectum of the ellipse, whose foci are (2,5) and (2,-3) and eccentricity is (4)/(5) , is

JEE Main 2025 Solution View → 24

Let C be the circle x^2+(y-1)^2=2, E_1 and E_2 be two ellipses whose centres lie at the origin and major axes lie on x -axis and y -axis respectively. Let the straight line x+y=3

JEE Main 2025 Solution View → 25

Let C be the circle of minimum area enclosing the ellipse E: (x^2)/(a^2)+(y^2)/(b^2)=1 with eccentricity (1)/(2) and foci ( ± 2,0) . Let PQR be a variable triangle, whose vertex P

JEE Main 2025 Solution View → 26

A line passing through the point P( 5 , 5 ) intersects the ellipse x^2 36 + y^2 25 =1 at A and B such that (P A) .(P B) is maximum. Then 5(P A^2+P B^2) is equal to :

JEE Main 2025 Solution View → 27

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 :

JEE Main 2025 Solution View → 28

If S and S^ are the foci of the ellipse (x^2)/(18)+(y^2)/(9)=1 and P be a point on the ellipse, then (S P . S^ P)+ (SP . S^ P) is equal to :

JEE Main 2025 Solution View → 29

If α x+β y=109 is the equation of the chord of the ellipse (x^2)/(9)+(y^2)/(4)=1 , whose mid point is ((5)/(2), (1)/(2)) , then α+β is equal to :

JEE Main 2025 Solution View → 30

Let the ellipse \( E _ 1 : x^2 a^2 + y^2 ~b^2 =1, a \) and \( E _ 2 : x^2 ~A^2 + y^2 ~B^2 = 1, ~A B \) have same eccentricity \( 1 3 \). Let the product of their lengths of latus

JEE Main 2025 Solution View → 31

If the midpoint of a chord of the ellipse (x^2)/(9)+(y^2)/(4)=1 is ( 2 , 4 / 3) , and the length of the chord is 2 α 3 , then α is :

JEE Main 2025 Solution View → 32

Let E_1: (x^2)/(9)+(y^2)/(4)=1 be an ellipse. Ellipses E_1 's are constructed such that their centres and eccentricities are same as that of E_1 , and the length of minor axis of

JEE Main 2025 Solution View → 33

The equation of the chord, of the ellipse (x^2)/(25)+(y^2)/(16)=1 , whose mid-point is (3,1) is :

JEE Main 2025 Solution View → 34

Let the product of the focal distances of the point ( 3 , (1)/(2)) on the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1,(a ) , be (7)/(4) . Then the absolute difference of the eccentricities

JEE Main 2025 Solution View → 35

The length of the chord of the ellipse (x^2)/(4)+(y^2)/(2)=1 , whose mid-point is (1, (1)/(2)) , is :

JEE Main 2025 Solution View → 36

Consider the ellipse (x^2)/(9)+(y^2)/(4)=1 . Let S(p, q) be a point in the first quadrant such that (p^2)/(9)+(q^2)/(4)>1 . Two tangents are drawn from S to the ellipse, of which

JEE Advanced 2024 Solution View → 37

Let f(x)=x^2+9, g(x)=(x)/(x-9) and a=f ∘ g(10), b=g ∘ f(3) . If e and l denote the eccentricity and the length of the latus rectum of the ellipse (x^2)/(a)+(y^2)/(b)=1 , then 8

JEE Main 2024 Solution View → 38

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 AB as a diameter is

JEE Main 2024 Solution View → 39

Let P be a point on the ellipse x ^ 2 9 + 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 PandQ are on

JEE Main 2024 Solution View → 40

Let P be a parabola with vertex (2, 3) and directrix 2x + y = 6 . Let an ellipse E: x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1, a > b of eccentricity 1 2 pass through the focus of the parabola

JEE Main 2024 Solution View → 41

Let A ( α, 0 ) and B ( 0, β ) be the points on the line 5x + 7y = 50 . Let the point P divide the line segment AB internally in the ratio 7: 3 . Let 3x - 25 = 0 be a directrix of

JEE Main 2024 Solution View → 42

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

JEE Main 2024 Solution View → 43

If the points of intersection of two distinct conics x ^ 2 + y ^ 2 = 4b and x ^ 2 16 + y ^ 2 b ^ 2 = 1 lie on the curve y ^ 2 = 3 x ^ 2 , then 3 3 times the area of the rectangle

JEE Main 2024 Solution View → 44

The length of the chord of the ellipse x ^ 2 25 + y ^ 2 16 = 1 , whose mid point is (1, (2)/(5)) , is equal to:

JEE Main 2024 Solution View → 45

Let T _ 1 and T _ 2 be two distinct common tangents to the ellipse E: x ^ 2 6 + y ^ 2 3 = 1 and the parabola P: y ^ 2 = 12x . Suppose that the tangent T _ 1 touches P and E at the

JEE Advanced 2023 Solution View → 46

Let an ellipse with centre (1, 0) and latus rectum of length (1)/(2) have its major axis along x-axis. If its minor axis subtends an angle 60 ^ ∘ at the foci, then the square of

JEE Main 2023 Solution View → 47

Let the tangent and normal at the point (3 3 , 1) on the ellipse x ^ 2 36 + y ^ 2 4 = 1 meet the y - axis at the points A and B respectively. Let the circle C be drawn taking AB

JEE Main 2023 Solution View → 48

Let P( 2 3 7 , 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 pass through the origin. If 1 (PQ) ^ 2 + 1 (RS)

JEE Main 2023 Solution View → 49

If the radius of the largest circle with centre ( 2, 0 ) inscribed in the ellipse x ^ 2 + 4 y ^ 2 = 36 is r , then 12 r ^ 2 is equal to

JEE Main 2023 Solution View → 50

Let a circle of radius 4 be concentric to the ellipse 15 x ^ 2 + 19 y ^ 2 = 285 . Then the common tangents are inclined to the minor axis of the ellipse at the angle

JEE Main 2023 Solution View → 51

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

JEE Main 2023 Solution View → 52

The line x = 8 is the directrix of the ellipse E: x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1 with the corresponding focus (2, 0) . If the tangent to E at the point P in the first quadrant

JEE Main 2023 Solution View → 53

If the tangent at a point P on the parabola y ^ 2 = 3x is parallel to the line x + 2y = 1 and the tangents at the points Q and R on the ellipse x ^ 2 4 + y ^ 2 1 = 1 are

JEE Main 2023 Solution View → 54

Let a tangent to the curve 9 x ^ 2 + 16 y ^ 2 = 144 intersect the coordinate axes at the points A and B . Then, the minimum length of the line segment AB is ______

JEE Main 2023 Solution View → 55

Let C be the largest circle centred at (2, 0) and inscribed in the ellipse x ^ 2 36 + y ^ 2 16 = 1 . If (1, α) lies on C , then 10 α ^ 2 is equal to ______

JEE Main 2023 Solution View → 56

Consider the ellipse x ^ 2 4 + y ^ 2 3 = 1 . Let H(α, 0), 0 List- I List- II (I) If ϕ = (π)/(4) , then the area of the triangle FGH is (P) ( 3 - 1) ^ 4 8 (II) If ϕ = (π)/(3)

JEE Advanced 2022 Solution View → 57

Let S = \ (x, y) ∈ ℕ × ℕ: 9 (x - 3) ^ 2 + 16 (y - 4) ^ 2 ≤ 144\ and T = \ (x, y) ∈ ℝ × ℝ: (x - 7) ^ 2 + (y - 4) ^ 2 ≤ 36\ The n(S ∩ T) is equal to ______.

JEE Main 2022 Solution View → 58

Let the tangents at the points P and Q on the ellipse x ^ 2 2 + y ^ 2 4 = 1 meet at the point R( 2 , 2 2 - 2) . If S is the focus of the ellipse on its negative major axis, then S

JEE Main 2022 Solution View → 59

If the length of the latus rectum of the ellipse x ^ 2 + 4 y ^ 2 + 2x + 8y - λ = 0 is 4 , and l is the length of its major axis, then λ + l is equal to _____.

JEE Main 2022 Solution View → 60

The acute angle between the pair of tangents drawn to the ellipse 2 x ^ 2 + 3 y ^ 2 = 5 from the point (1, 3) is

JEE Main 2022 Solution View → 61

If the ellipse x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1 meets the line (x)/(7) + y 2 6 = 1 on the x -axis and the line (x)/(7) - y 2 6 = 1 on the y -axis, then the eccentricity of the

JEE Main 2022 Solution View → 62

Let PQ be a focal chord of the parabola y ^ 2 = 4x such that it subtends an angle of (π)/(2) at the point (3, 0) . Let the line segment PQ be also a focal chord of the ellipse E

JEE Main 2022 Solution View → 63

Let the eccentricity of an ellipse x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1, a > b , be (1)/(4) . If this ellipse passes through the point (- 4 (2)/(5) , 3) , then a ^ 2 + b ^ 2 is equal to

JEE Main 2022 Solution View → 64

If m is the slope of a common tangent to the curves x ^ 2 16 + y ^ 2 9 = 1 and x ^ 2 + y ^ 2 = 12 , then 12 m ^ 2 is equal to

JEE Main 2022 Solution View → 65

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

JEE Main 2022 Solution View → 66

The line y = x + 1 meets the ellipse x ^ 2 4 + 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

JEE Main 2022 Solution View → 67

Let the maximum area of the triangle that can be inscribed in the ellipse x ^ 2 a ^ 2 + y ^ 2 4 = 1, a > 2 , having one of its vertices at one end of the major axis of the ellipse

JEE Main 2022 Solution View → 68

Let E be the ellipse x ^ 2 16 + y ^ 2 9 = 1 . For any three distinct points P, Q and Q ^ ' on E , let M(P, Q) be the mid-point of the line segment joining P and Q , and M(P, Q ^ '

JEE Advanced 2021 Solution View → 69

Let θ be the acute angle between the tangents to the ellipse x ^ 2 9 + y ^ 2 1 = 1 and the circle x ^ 2 + y ^ 2 = 3 at their point of intersection in the first quadrant. Then tanθ

JEE Main 2021 Solution View → 70

The locus of mid-points of the line segments joining (- 3, - 5) and the points on the ellipse x ^ 2 4 + y ^ 2 9 = 1 is :

JEE Main 2021 Solution View → 71

The line 12xcosθ + 5ysinθ = 60 is tangent to which of the following curves ?

JEE Main 2021 Solution View → 72

If the minimum area of the triangle formed by a tangent to the ellipse x ^ 2 b ^ 2 + y ^ 2 4 a ^ 2 = 1 and the co-ordinate axis is kab, then k is equal to ___________.

JEE Main 2021 Solution View → 73

On the ellipse x ^ 2 8 + y ^ 2 4 = 1 , let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0 . Let S and S ^ '

JEE Main 2021 Solution View → 74

Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its centre at (3, - 4), one focus at (4, - 4) and one vertex at (5, - 4) . If mx - y = 4, m > 0 is a

JEE Main 2021 Solution View → 75

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

JEE Main 2021 Solution View → 76

If a tangent to the ellipse x ^ 2 + 4 y ^ 2 = 4 meets the tangents at the extremities of its major axis at B and C, then the circle with BC as diameter passes through the point.

JEE Main 2021 Solution View → 77

Let an ellipse E: x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1, a ^ 2 > b ^ 2 , passes through ( (3)/(2) , 1) and has eccentricity 1 3 . If a circle, centered at focus F ( α, 0 ) , α > 0 , of E

JEE Main 2021 Solution View → 78

Let E _ 1 : x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1, a > b . Let E _ 2 be another ellipse such that it touches the end points of major axis of E _ 1 and the foci of E _ 2 are the end points

JEE Main 2021 Solution View → 79

Let a tangent be drawn to the ellipse x ^ 2 27 + y ^ 2 = 1 at ( 3 3 cosθ, sinθ ) where θ ∈ (0, (π)/(2)) . Then the value of θ such that the sum of intercepts on axes made by this

JEE Main 2021 Solution View → 80

Let L be a tangent line to the parabola y ^ 2 = 4x - 20 at (6, 2) . If L is also a tangent to the ellipse x ^ 2 2 + y ^ 2 b = 1, then the value of b is equal to :

JEE Main 2021 Solution View → 81

If the points of intersection of the ellipse x ^ 2 16 + y ^ 2 b ^ 2 = 1 and the circle x ^ 2 + y ^ 2 = 4b, b > 4 lie on the curve y ^ 2 = 3 x ^ 2 , then b is equal to :

JEE Main 2021 Solution View → 82

Let L be a common tangent line to the curves 4 x ^ 2 + 9 y ^ 2 = 36 and (2x) ^ 2 + (2y) ^ 2 = 31 . Then the square of the slope of the line L is ______.

JEE Main 2021 Solution View → 83

If the curve x ^ 2 + 2 y ^ 2 = 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

JEE Main 2021 Solution View → 84

If the curves, x ^ 2 a + y ^ 2 b = 1 and x ^ 2 c + y ^ 2 d = 1 intersect each other at an angle of 90 ^ ° , then which of the following relations is TRUE?

JEE Main 2021 Solution View → 85

If the normal at an end of latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies :

JEE Main 2020 Solution View → 86

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, x ^ 2 4 + y ^ 2 2 = 1 from any of its foci?

JEE Main 2020 Solution View → 87

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, 9 x ^ 2 + 16 y ^ 2 = 144, then PA + PB is equal to :

JEE Main 2020 Solution View → 88

If the point P on the curve, 4 x ^ 2 + 5 y ^ 2 = 20 is farthest from the point Q(0, - 4) , then P Q ^ 2 is equal to

JEE Main 2020 Solution View → 89

Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is (1)/(2) . If P ( 1, β ) , β > 0 is a point on this ellipse, then the equation of the

JEE Main 2020 Solution View → 90

Let x ^ 2 a ^ 2 + 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, ϕ(t) = (5)/(12) + t - t

JEE Main 2020 Solution View → 91

The length of the minor axis (along y -axis) of an ellipse in the standard form is 4 3 . If this ellipse touches the line x + 6y = 8 then its eccentricity is:

JEE Main 2020 Solution View → 92

Let the line y = mx and the ellipse 2 x ^ 2 + y ^ 2 = 1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at (- 1 3 2

JEE Main 2020 Solution View → 93

If 3x + 4y = 12 2 is a tangent o the ellipse x ^ 2 a ^ 2 + y ^ 2 9 = 1 for some a ∈ R, then the distance between the foci of the ellipse is

JEE Main 2020 Solution View → 94

If the distance between the foci of an ellipse is 6 and the distance between its directrix is 12, then the length of its latus rectum is

JEE Main 2020 Solution View → 95

The angle between the pair of tangents drawn to the ellipse 3 x ^ 2 + 2 y ^ 2 = 5 from the point (1, 2) is | ^ - 1 ( 12 λ )|, then the value of λ is

NTA Abhyas JEE Main 2020 Solution View → 96

If the line x - 2y = 12 is a tangent to the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 = 1 at the point (3, - (9)/(2)) , then the length of the latus rectum of the ellipse is

NTA Abhyas JEE Main 2020 Solution View → 97

The point, which is at the shortest distance from the line x + y = 7 and lying on an ellipse x ^ 2 + 2 y ^ 2 = 6 , has coordinates ( a, b ) then the value of (a)/(b) is

NTA Abhyas JEE Main 2020 Solution View → 98

If the locus of the foot of the perpendicular drawn from centre upon any tangent to the ellipse x^ 2 40 + y^ 2 10 =1 is (x^ 2 + y^ 2 ) ^ 2 = ax^ 2 + by^ 2 , then (a – b) is equal

NTA Abhyas JEE Main 2020 Solution View → 99

If common tangents of x ^ 2 + y ^ 2 = r ^ 2 and x ^ 2 16 + y ^ 2 9 = 1 forms a square, then the length of diagonal of the square is

NTA Abhyas JEE Main 2020 Solution View → 100

If a tangent of slope 2 of the ellipse x ^ 2 a ^ 2 + y ^ 2 1 = 1 passes through the point (- 2,0) , then the value of a ^ 2 is equal to

NTA Abhyas JEE Main 2020 Solution View → 101

Let from a point A (h, k) chord of contacts are drawn to the ellipse x^ 2 + 2y^ 2 = 6 such that all these chords touch the ellipse x^ 2 + 4y^ 2 = 4 , then locus of the point A is

NTA Abhyas JEE Main 2020 Solution View → 102

If any tangent to the ellipse 25x^ 2 + 9y^ 2 = 225 meets the coordinate axes at A and B such that OA = OB then, the length AB is equal to (where, O is the origin)

NTA Abhyas JEE Main 2020 Solution View → 103

If the normal to the ellipse x ^ 2 25 + y ^ 2 1 = 1 is at a distance p from the origin, then the maximum value of p is

NTA Abhyas JEE Main 2020 Solution View → 104

Suppose S and S′ are foci of the ellipse x^ 2 25 + y^ 2 16 = 1 . If P is a variable point on the ellipse and if ∆ is the area (in sq. units) of the triangle PSS′ , then the

NTA Abhyas JEE Main 2020 Solution View → 105

The line 2x + y = 3 cuts the ellipse 4 x ^ 2 + y ^ 2 = 5 at points P and Q . If θ is the acute angle between the normals at P and Q , then θ is equal to

NTA Abhyas JEE Main 2020 Solution View → 106

A tangent having slope& - (4)/(3) to the ellipse x^ 2 18 + y^ 2 32 = 1 intersects the major and minor axes at A and B . If O is the origin, then the area of Δ OAB is

NTA Abhyas JEE Main 2020 Solution View → 107

An ellipse has foci (4,2), (2,2) and it passes through the pointP (2,4) . The eccentricity of the ellipse is

NTA Abhyas JEE Main 2020 Solution View → 108

The line 2x + y = 3 interesects the ellipse 4x^ 2 + y^ 2 = 5 at two points. The point of intersection of the tangents to the ellipse at these points is

NTA Abhyas JEE Main 2020 Solution View → 109

Let the line y = mx and the ellipse 2x^ 2 + y^ 2 = 1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at (- 1 3 2 , 0)

NTA Abhyas JEE Main 2020 Solution View → 110

The length of the latus rectum of the curve represented by x = 3 (cos ⁡ t + sin ⁡ t) and y = 4(cos ⁡ t - sin ⁡ t), is

NTA Abhyas JEE Main 2020 Solution View → 111

If m _ 1 ~and& m _ 2 are the slopes of the tangents to the ellipse x^ 2 16 + y^ 2 9 = 1 which passes through (5,4), ~then the value of ( m _ 1 + m _ 2 ) - ( m _ 1 m _ 2 ) is equal

NTA Abhyas JEE Main 2020 Solution View → 112

If P and Q are points with eccentric angles θ and (θ + (π)/(6)) on the ellipse x ^ 2 16 + y ^ 2 4 = 1, then the area (in sq. units) of the triangle OPQ (where O is the origin) is

NTA Abhyas JEE Main 2020 Solution View → 113

If y = x + c touches the ellipse 3x^ 2 + 4y^ 2 = 12 at the point P, then the value of the length OP (where O is the origin) is equal to

NTA Abhyas JEE Main 2020 Solution View → 114

If θ is the angle between the pair of tangents drawn to the ellipse 3 x ^ 2 + 2 y ^ 2 = 5 from the point (1,2), then the value of ^ 2 θ is equal to

NTA Abhyas JEE Main 2020 Solution View → 115

The focus and corresponding directrix of an ellipse are (3,4) and x + y - 1 = 0 respectively. If the eccentricity of the ellipse is (1)/(2) , then the coordinates of the centre of

NTA Abhyas JEE Main 2020 Solution View → 116

If (0,3 + 5 ) is a point on the ellipse whose foci are (2,3) and (- 2,3), then the length of the semi-major axis is

NTA Abhyas JEE Main 2020 Solution View → 117

Let S and S^ ′ are the focii of the ellipse x = 3 + 5cos ⁡ θ, y = - 2 + 4sin ⁡ θ. If B is one of the ends of one of the latus rectum, then the area (in sq. units) of the triangle

NTA Abhyas JEE Main 2020 Solution View → 118

The length of the portion of the common tangent to x^ 2 + y^ 2 = 16 and 9x^ 2 + 25y^ 2 = 225 & between the two points of contact is

NTA Abhyas JEE Main 2020 Solution View → 119

Let there are exactly two points on the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 = 1 whose distances from (0,0) are equal to a^ 2 2 + b^ 2 . Then, the eccentricity of the ellipse is equal to

NTA Abhyas JEE Main 2020 Solution View → 120

The point on the ellipse x^ 2 6 + y^ 2 3 = 1 which is nearest to the line x + y = 7 is

NTA Abhyas JEE Main 2020 Solution View → 121

P _ 1 and P _ 2 are corresponding points on the ellipse x^ 2 16 + y^ 2 9 = 1 and its auxiliary circle respectively. If the normal at P _ 1 to the ellipse meets O P _ 2 in Q (where

NTA Abhyas JEE Main 2020 Solution View → 122

If two points are taken on the minor axis of the ellipse x^ 2 25 + y^ 2 9 = 1 at the same distance from the centre as the foci, then the sum of the squares of the perpendicular

NTA Abhyas JEE Main 2020 Solution View → 123

The length of the minor axis of the ellipse (5x - 10) ^ 2 + (5y + 15) ^ 2 = (3x - 4y + 7) ^ 2 4 is

NTA Abhyas JEE Main 2020 Solution View → 124

If a tangent having slope 2 of the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 = 1 is normal to the circle x^ 2 + y^ 2 + 4x + 1 = 0, then the value of 4a^ 2 + b^ 2 is equal to

NTA Abhyas JEE Main 2020 Solution View → 125

Variable ellipses are drawn with x = - 4 as directrix and the origin as the corresponding focus. The eccentricity of the locus of the extremities of the minor axes of these

NTA Abhyas JEE Main 2020 Solution View → 126

The tangents at the extremities of the latus rectum of the ellipse 3x^ 2 + 4y^ 2 = 12 form a rhombus PQRS . Area (in sq. units) of the rhombus PQRS outside the ellipse is equal to

NTA Abhyas JEE Main 2020 Solution View → 127

The ellipse E _ 1 : x ^ 2 9 + y ^ 2 4 = 1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E _ 2 passing through the point (0, 4)

NTA Abhyas JEE Main 2020 Solution View → 128

An equilateral triangle is inscribed in the ellipse whose equation is x^ 2 + 4y^ 2 = 4 . One vertex of the triangle is (0,1) and one altitude is contained in the~ y -axis. If the

NTA Abhyas JEE Main 2020 Solution View → 129

If the tangent to the ellipse x^ 2 + 4y^ 2 = 16 at P (4cos ⁡ θ,2sin ⁡ θ) is also a normal to the circle x^ 2 + y^ 2 = 8x + 4y,then the sum of all possible values of θ is equal to

NTA Abhyas JEE Main 2020 Solution View → 130

If the circle whose diameter is the major axis of the ellipse (x2)/(a2) + (y2)/(b2) = 1 (a > b > 0) meets the minor axis at point P and the orthocentre of ∆ P F _ 1 F _ 2 lies on

NTA Abhyas JEE Main 2020 Solution View → 131

The product of the slopes of the common tangents of the& ellipse x^ 2 + 4y^ 2 = 16 and the parabola y^ 2 - 4x - 4 = 0 & is

NTA Abhyas JEE Main 2020 Solution View → 132

If x^ 2 + y^ 2 = a^ 2 and x^ 2 16 + y^ 2 9 = 1 intersect at 4 points P, Q, R and S which form a square, then the area (in sq. units) of the square is

NTA Abhyas JEE Main 2020 Solution View → 133

A tangent of slope 2 of the ellipse x ^ 2 a ^ 2 + y ^ 2 1 = 1 passes through (- 2, 0) . Then, three times the square of the eccentricity of the ellipse is equal to

NTA Abhyas JEE Main 2020 Solution View → 134

Let from a point A (h, k) chords of contact are drawn to the ellipse x^ 2 + 2y^ 2 = 6 where all these chords touch the ellipse x^ 2 + 4y^ 2 = 4 . Then, the perimeter (in units) of

NTA Abhyas JEE Main 2020 Solution View → 135

If the tangents PQ and PR are drawn from a variable point P to the ellipse x^ 2 16 + y^ 2 9 = 1, such that the fourth point S of the parallelogram PQSR lies on the circumcircle of

NTA Abhyas JEE Main 2020 Solution View → 136

The line 2x + y = 3 cuts the ellipse 4 x ^ 2 + y ^ 2 = 5 at P and Q . If θ is the acute angle between normal at P and tangent at Q , then θ is equal to

NTA Abhyas JEE Main 2020 Solution View → 137

If the common tangents of x^ 2 + y^ 2 = r^ 2 and x^ 2 16 + y^ 2 9 = 1 form a square, then the area (in sq. units) of the square is

NTA Abhyas JEE Main 2020 Solution View → 138

If eccentricity of the ellipse x^ 2 a^ 2 + 1 + y^ 2 a^ 2 + 2 = 1 is 1 6 , then the ratio of the length of the latus rectum to the length of the major axis is

NTA Abhyas JEE Main 2020 Solution View → 139

If a, b ∈ R satisfy the equation a^ 2 + 4b^ 2 - 4 = 0, then the minimum value of (2a + 3b) will be

NTA Abhyas JEE Main 2020 Solution View → 140

Two lines L _ 1 and L _ 2 of slope 1 are tangent to y^ 2 = 4x and x^ 2 + 2y^ 2 = 4 respectively, such that the distance d units between L _ 1 and L _ 2 is minimum, then the value

NTA Abhyas JEE Main 2020 Solution View → 141

The focal chord of the parabola y^ 2 = 32x touches the ellipse x^ 2 4 ^ 2 + y^ 2 2 ^ 2 = 1 in the first quadrant at the point

NTA Abhyas JEE Main 2020 Solution View → 142

A trapezium is formed by the pair of tangents of parabola P: y = x^ 2 4 + 1 drawn from the centre of the ellipse E: x^ 2 4 + y^ 2 = (1)/(4) , tangent at the vertex of P and the

NTA Abhyas JEE Main 2020 Solution View → 143

From the point P (3,4) pair of tangents PA and PB are drawn to the ellipse x^ 2 16 + y^ 2 9 =1. If AB intersects y -axis at C and x -axis at D , then OC is equal to (where O is

NTA Abhyas JEE Main 2020 Solution View → 144

The tangent to the ellipse x^ 2 25 + y^ 2 16 = 1 at point P lying in the first quadrant meets x -axis at Q and y -axis at R . If the length QR is minimum, then the equation of

NTA Abhyas JEE Main 2020 Solution View → 145

The vertices of the triangle ABC are A(0,0), B(3, 0) and C(3, 4) , where A and C are foci of an ellipse and B lies on the ellipse. If the length of the latus rectum of the ellipse

NTA Abhyas JEE Main 2020 Solution View → 146

If a number of ellipses whose major axis is x -axis and the minor axis is y -axis be described having the same length of the major axis as 2 but a variable minor axis, then the

NTA Abhyas JEE Main 2020 Solution View → 147

A normal is drawn to the ellipse (x2)/(9) + y^ 2 = 1 at the point (3cos ⁡ θ, sin ⁡ θ) where 0 < θ < (π)/(2) . If N is the foot of the perpendicular from the origin~ O ~to the

NTA Abhyas JEE Main 2020 Solution View → 148

Define the collections \ E _ 1 , E _ 2 , E _ 3 , . . . .\ of ellipses and \ R _ 1 , R _ 2 , R _ 3 , . . . .\ of rectangles as follows: E _ 1 : x ^ 2 9 + y ^ 2 4 = 1; R _ 1

JEE Advanced 2019 Solution View → 149

An ellipse, with foci at (0,2) and (0, - 2) and minor axis of length 4 , passes through which of the following points?

JEE Main 2019 Solution View → 150

If the normal to the ellipse 3x ^ 2 + 4 y ^ 2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to the ellipse at P passes through Q ( 4,4 ) then PQ is

JEE Main 2019 Solution View → 151

The tangent and normal to the ellipse 3 x ^ 2 + 5 y ^ 2 = 32 at the point P(2, 2) meet the x -axis at Q and R , respectively. Then the area (in sq. units) of the triangle PQR is:

JEE Main 2019 Solution View → 152

If the line x - 2y = 12 is a tangent to the ellipse x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1 at the point (3, - (9)/(2)) , then the length of the latus rectum of the ellipse is

JEE Main 2019 Solution View → 153

If the tangent to the parabola y ^ 2 = x at a point (α, β), (β > 0) is also a tangent to the ellipse, x ^ 2 + 2 y ^ 2 = 1 then α is equal to:

JEE Main 2019 Solution View → 154

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

JEE Main 2019 Solution View → 155

Let O(0,0) and A(0,1) be two fixed points. Then, the locus of a point P such that the perimeter of is 4 is

JEE Main 2019 Solution View → 156

If the tangents on the ellipse 4 x ^ 2 + y ^ 2 = 8 at the points (1,2) and ( a, b ) are perpendicular to each other, then a ^ 2 is equal to

JEE Main 2019 Solution View → 157

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 ^ '

JEE Main 2019 Solution View → 158

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

JEE Main 2019 Solution View → 159

If tangents are drawn to the ellipse x^ 2 +2 y^ 2 =2 at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between the

JEE Main 2019 Solution View → 160

Let S = \ (x, y) ∈ R ^ 2 : y ^ 2 1 + r - x ^ 2 1 - r = 1\ , where r ≠ ± 1 . Then S represents:

JEE Main 2019 Solution View → 161

Consider two straight lines, each of which is tangent to both the circle x ^ 2 + y ^ 2 = (1)/(2) and the parabola y ^ 2 = 4x . Let these lines intersect at the point Q . Consider

JEE Advanced 2018 Solution View → 162

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is (3)/(2) units, then its eccentricity is

JEE Main 2018 Solution View → 163

If β is one of the angles between the normals to the ellipse, x^2+3 y^2=9 at the points (3 θ, 3 θ) and (-3 θ, 3 θ) ; ∈(0, (π)/(2)) ; then (2 β)/( 2 θ) is equal to

JEE Main 2018 Solution View → 164

If β is one of the angles between the normals to the ellipse x ^ 2 + 3 y ^ 2 = 9 at the points (3cosθ, 3 sinθ) and (- 3sinθ, 3 cosθ) ; θ ∈ (0, (π)/(2)) ; then (2cotβ)/(sin2θ) is

JEE Main 2018 Solution View → 165

Two sets A and B are as under: A = \ (a, b) ∈ R × R: |a - 5| B = \ (a, b) ∈ R × R: 4 (a - 6) ^ 2 + 9 (b - 5) ^ 2 ≤ 36\ . Then :

JEE Main 2018 Solution View → 166

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. Column 1 Column 2 Column 3 (I) x ^ 2 + y ^ 2 = a ^ 2 (i) my = m ^ 2 x +

JEE Advanced 2017 Solution View → 167

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively. Column 1 Column 2 Column 3 (I) x ^ 2 + y ^ 2 = a ^ 2 (i) my = m ^ 2 x +

JEE Advanced 2017 Solution View → 168

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

JEE Main 2017 Solution View → 169

Consider an ellipse, whose center is at the origin and its major axis is along the x -axis. If its eccentricity is (3)/(5) and the distance between its foci is 6 , then the area

JEE Main 2017 Solution View → 170

The eccentricity of an ellipse whose centre is at the origin is (1)/(2) . If one of its directrices is x = - 4 , then the equation of the normal to it at (1, (3)/(2)) is:

JEE Main 2017 Solution View → 171

Paragraph: Let F_ 1 (x_ 1 , 0) and F_ 2 (x_ 2 , 0) , for x_ 1 0 , be the foci of the ellipse x^ 2 9 + y^ 2 8 =1 . Suppose a parabola having vertex at the origin and focus at F_ 2

JEE Advanced 2016 Solution View → 172

If the tangent at a point on the ellipse x ^ 2 27 + y ^ 2 3 = 1 meets the coordinate axes at A and B , and O is the origin, then the minimum area (in sq. units) of the triangle

JEE Main 2016 Solution View → 173

Let E _ 1 and E _ 2 be two ellipse whose centers are at the origin. The major axes of E _ 1 and E _ 2 lie along the x - axis and the y - axis, respectively. Let S be the circle x

JEE Advanced 2015 Solution View → 174

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is:

JEE Main 2015 Solution View → 175

An ellipse passes through the foci of the hyperbola, 9 x ^ 2 - 4 y ^ 2 = 36 and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively.

JEE Main 2015 Solution View → 176

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latus ractum to the ellipse x ^ 2 9 + y ^ 2 5 = 1, is

JEE Main 2015 Solution View → 177

The minimum area of a triangle formed by any tangent to the ellipse (x^2)/(16)+(y^2)/(81)=1 and the co-ordinate axes is:

JEE Main 2014 Solution View → 178

A stair-case of length l rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the

JEE Main 2014 Solution View → 179

If OB is the semi-minor axis of an ellipse, F _ 1 and F _ 2 are its focii and the angle between F _ 1 B and F _ 2 B is a right angle, then the square of the eccentricity of the

JEE Main 2014 Solution View → 180

A vertical line passing through the point (h , 0) intersects the ellipse x ^ 2 4 + y ^ 2 3 = 1at the points P and Q . Let the tangents to the ellipse at P and Q meet at the point

JEE Advanced 2013 Solution View → 181

A point on the ellipse, 4 x^2+9 y^2=36 , where the normal is parallel to the line, 4 x-2 y-5=0 , is :

JEE Main 2013 Solution View → 182

Let the equations of two ellipses be E_1: (x^2)/(3)+(y^2)/(2)= 1 and E_2: (x^2)/(16)+(y^2)/(b^2)= 1 , If the product of their eccentricities is (1)/(2) , then the length of the

JEE Main 2013 Solution View → 183

Equation of the line passing through the points of intersection of the parabola x^2=8 y and the ellipse (x^2)/(3)+y^2=1 is :

JEE Main 2013 Solution View → 184

If a and c are positive real numbers and the ellipse (x^2)/(4 c^2)+(y^2)/(c^2)=1 has four distinct points ir common with the circle x^2+y^2=9 a^2 , then

JEE Main 2013 Solution View → 185

The equation of the circle passing through the foci of the ellipse x ^ 2 16 + y ^ 2 9 = 1 , and having centre at (0, 3) is

JEE Main 2013 Solution View → 186

The ellipse E_ 1 : x^ 2 9 + y^ 2 4 =1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E_ 2 passing through the point (0,4)

JEE Advanced 2012 Solution View → 187

The normal at (2, (3)/(2)) to the ellipse, (x^2)/(16)+(y^2)/(3)=1 touches a parabola, whose equation is

JEE Main 2012 Solution View → 188

If the foci of the ellipse (x^2)/(16)+(y^2)/(b^2)=1 coincide with the foci of the hyperbola (x^2)/(144)-(y^2)/(81)=(1)/(25) , then b^2 is equal to

JEE Main 2012 Solution View → 189

If P_1 and P_2 are two points on the ellipse (x^2)/(4)+y^2=1 at which the tangents are parallel to the chord joining the points (0,1) and (2,0) , then the distance between P_1 and

JEE Main 2012 Solution View → 190

An ellipse is drawn by taking a diameter of the circle (x-1)^2+y^2=1 as its semiminor axis and a diameter of the circle x^2+(y-2)^2=4 as its semi-major axis. If the centre of the

JEE Main 2012 Solution View → 191

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (-3,1) and has eccentricity (2)/(5) is

JEE Main 2011 Solution View → 192

Paragraph: Tangents are drawn from the point P(3,4) to the ellipse (x^2)/(9)+(y^2)/(4)=1 touching the ellipse at points A and B . Question: The coordinates of A and B are

JEE Advanced 2010 Solution View → 193

Paragraph: Tangents are drawn from the point P(3,4) to the ellipse (x^2)/(9)+(y^2)/(4)=1 touching the ellipse at points A and B . Question: The equation of the locus of the point

JEE Advanced 2010 Solution View → 194

The normal at a point P on the ellipse x^2+4 y^2=16 meets the x -axis at Q . If M is the mid-point of the line segment P Q , then the locus of M intersects the latusrectum of the

JEE Advanced 2009 Solution View → 195

An ellipse intersects the hyperbola 2 x^2-2 y^2=1 orthogonally. The eccentricity of the ellipse is reciprocal to that of the hyperbola. If the axes of the ellipse are along the

JEE Advanced 2009 Solution View → 196

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x^2+9 y^2=9 meets its auxiliary circle at the point M . Then, the area

JEE Advanced 2009 Solution View → 197

The ellipse x^2+4 y^2=4 is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point (4,0) . Then the

JEE Main 2009 Solution View → 198

Let P(x_1, y_1) and Q(x_2, y_2), y_1 < 0, y_2 < 0 , be the end points of the latusrectum of the ellipse x^2+4 y^2=4 . The equations of parabolas with latusrectum P Q are

JEE Advanced 2008 Solution View → 199

A focus of an ellipse is at the origin. The directrix is the line x=4 and the eccentricity is 1 / 2 . Then the length of the semi-major axis is

JEE Main 2008 Solution View → 200

A hyperbola, having the transverse axis of length 2 θ , is confocal with the ellipse 3 x^2+4 y^2=12 . Then, its equation is

JEE Advanced 2007 Solution View →
Practice all Ellipse questions in the app →

Related JEE Mathematics chapters

Application of Derivatives (795)Three Dimensional Geometry (756)Sequences and Series (748)Functions (700)Definite Integration (692)Vector Algebra (616)Differential Equations (586)Continuity and Differentiability (576)Probability (561)Complex Number (551)Binomial Theorem (550)Quadratic Equation (517)

Ellipse in other exams

NDA Ellipse (38)BITSAT Ellipse (12)