Conic Sections

39 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q11·MCQ·Easy–moderate
Open
Hyperbola latus rectum
Let the foci of hyperbola coincide with the foci of the ellipse $\dfrac{x^2}{36} + \dfrac{y^2}{16} = 1$ . If the eccentricity of the hyperbola is 5, then the length of its latus rectum is:
1. (1)12
2. (2)16
3. (3) $\dfrac{96}{\sqrt5}$
4. (4) $24\sqrt5$
MathsJEE Main 21 Jan 2026, Shift 1·Q20·MCQ·Moderate
Open
Parabola chord with given midpoint
Let $O$ be the vertex of the parabola $x^2 = 4y$ and $Q$ be any point on it. Let the locus of the point $P$ , which divides the line segment $OQ$ internally in the ratio $2:3$ be the conic $C$ . Then the equation of the chord of $C$ , which is bisected at the point $(1, 2)$ , is:
1. (1) $5x - y - 3 = 0$
2. (2) $4x - 5y + 6 = 0$
3. (3) $x - 2y + 3 = 0$
4. (4) $5x - 4y + 3 = 0$
We corrected a printing error in the original paper
Source labels the options “(1), (2), (4), (4)”; the third option ( $x - 2y + 3 = 0$ ) is printed as “(4)”, the key (4) is the last option.
MathsJEE Main 21 Jan 2026, Shift 2·Q4·MCQ·Moderate
Open
Ellipse tangent and focal distance
If the line $\alpha x + 4y = \sqrt7$ , where $\alpha \in \mathbb{R}$ , touches the ellipse $3x^2 + 4y^2 = 1$ at the point $P$ in the first quadrant, then one of the focal distances of $P$ is:
1. (1) $\dfrac{1}{\sqrt3} - \dfrac{1}{2\sqrt{11}}$
2. (2) $\dfrac{1}{\sqrt3} + \dfrac{1}{2\sqrt5}$
3. (3) $\dfrac{1}{\sqrt3} - \dfrac{1}{2\sqrt5}$
4. (4) $\dfrac{1}{\sqrt3} + \dfrac{1}{2\sqrt7}$
We corrected a printing error in the original paper
Source printed “In the line” for “If the line”.
MathsJEE Main 21 Jan 2026, Shift 2·Q5·MCQ·Moderate
Open
Locus of centroid on parabola
Let $y^2 = 12x$ be the parabola with its vertex at $O$ . Let $P$ be a point on the parabola and $A$ be a point on the $x$ -axis such that $\angle OPA = 90^\circ$ . Then the locus of the centroid of such triangles $OPA$ is:
1. (1) $y^2 - 6x + 4 = 0$
2. (2) $y^2 - 9x + 6 = 0$
3. (3) $y^2 - 2x + 8 = 0$
4. (4) $y^2 - 4x + 8 = 0$
MathsJEE Main 21 Jan 2026, Shift 2·Q6·MCQ·Easy–moderate
Open
Focal chord of a parabola
Let one end of a focal chord of the parabola $y^2 = 16x$ be $(16, 16)$ . If $P(\alpha, \beta)$ divides this focal chord internally in the ratio $5:2$ , then the minimum value of $\alpha + \beta$ is equal to:
1. (1)22
2. (2)7
3. (3)5
4. (4)16
MathsJEE Main 22 Jan 2026, Shift 1·Q10·MCQ·Moderate
Open
Line not meeting a hyperbola
If the line $\alpha x + 2y = 1$ , where $\alpha \in \mathbb{R}$ , does not meet the hyperbola $x^2 - 9y^2 = 9$ , then a possible value of $\alpha$ is:
1. (1)0.6
2. (2)0.8
3. (3)0.5
4. (4)0.7
MathsJEE Main 22 Jan 2026, Shift 1·Q17·MCQ·Easy–moderate
Open
Parabola chord subtending right angle
If the chord joining the points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ on the parabola $y^2 = 12x$ subtends a right angle at the vertex of the parabola, then $x_1x_2 - y_1y_2$ is equal to
1. (1)288
2. (2)280
3. (3)284
4. (4)292
MathsJEE Main 22 Jan 2026, Shift 2·Q11·MCQ·Moderate
Open
Hyperbola latus rectum and foci
Let $P\left(10, 2\sqrt{15}\right)$ be a point on the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ , whose foci are $S$ and $S'$ . If the length of its latus rectum is 8, then the square of the area of $\Delta PSS'$ is equal to:
1. (1)4200
2. (2)900
3. (3)1462
4. (4)2700
MathsJEE Main 22 Jan 2026, Shift 2·Q16·MCQ·Moderate
Open
Ellipse focal distances
Let $S$ and $S'$ be the foci of the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ and $P(\alpha, \beta)$ be a point on the ellipse in the first quadrant. If $(SP)^2 + (S'P)^2 - SP\cdot S'P = 37$ , then $\alpha^2 + \beta^2$ is equal to:
1. (1)15
2. (2)11
3. (3)17
4. (4)13
MathsJEE Main 22 Jan 2026, Shift 2·Q17·MCQ·Moderate
Open
Locus of chord mid-points
Let the locus of the mid-point of the chord through the origin $O$ of the parabola $y^2 = 4x$ be the curve $S$ . Let $P$ be any point on $S$ . Then the locus of the point, which internally divides $OP$ in the ratio $3:1$ , is:
1. (1) $3y^2 = 2x$
2. (2) $2y^2 = 3x$
3. (3) $3x^2 = 2y$
4. (4) $2x^2 = 3y$
MathsJEE Main 23 Jan 2026, Shift 1·Q5·MCQ·Moderate
Open
Ellipse and line intersection
Let the line $y - x = 1$ intersect the ellipse $\dfrac{x^2}{2} + \dfrac{y^2}{1} = 1$ at the points $A$ and $B$ . Then the angle made by the line segment $AB$ at the center of the ellipse is:
1. (1) $\pi - \tan^{-1}\left(\dfrac14\right)$
2. (2) $\dfrac{\pi}{2} + \tan^{-1}\left(\dfrac14\right)$
3. (3) $\dfrac{\pi}{2} + 2\tan^{-1}\left(\dfrac14\right)$
4. (4) $\dfrac{\pi}{2} - \tan^{-1}\left(\dfrac14\right)$
MathsJEE Main 23 Jan 2026, Shift 2·Q8·MCQ·Moderate
Open
Curves through intersection points
If the points of intersection of the ellipses $x^2 + 2y^2 - 6x - 12y + 23 = 0$ and $4x^2 + 2y^2 - 20x - 12y + 35 = 0$ lie on a circle of radius $r$ and centre $(a, b)$ , then the value of $ab + 18r^2$ is
1. (1)53
2. (2)51
3. (3)52
4. (4)55
We corrected a printing error in the original paper
Source printed “ $4x + 2y^2 - 20x - 12y + 35 = 0$ ”; corrected to the intended “ $4x^2 + 2y^2 - 20x - 12y + 35 = 0$ ” (as printed the second curve is a parabola, not an ellipse, and meets the first in only two points, so the circle is not determined).
MathsJEE Main 23 Jan 2026, Shift 2·Q9·MCQ·Moderate
Open
Hyperbola chord and eccentricity
Let $PQ$ be a chord of the hyperbola $\dfrac{x^2}{4} - \dfrac{y^2}{b^2} = 1$ , perpendicular to the $x$ -axis such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola. If the eccentricity of the hyperbola is $\sqrt3$ , then the area of the triangle $OPQ$ is:
1. (1) $2\sqrt3$
2. (2) $\dfrac{8\sqrt3}{5}$
3. (3) $\dfrac{11}{5}$
4. (4) $\dfrac{9}{5}$
MathsJEE Main 23 Jan 2026, Shift 2·Q19·MCQ·Moderate
Open
Equilateral triangle in parabola
An equilateral triangle $OAB$ is inscribed in the parabola $y^2 = 4x$ with the vertex $O$ at the vertex of the parabola. Then the minimum distance of the circle having $AB$ as a diameter from the origin is
1. (1) $4(3 - \sqrt3)$
2. (2) $2(8 - 3\sqrt3)$
3. (3) $4(6 + \sqrt3)$
4. (4) $2(3 + \sqrt3)$
MathsJEE Main 24 Jan 2026, Shift 1·Q7·MCQ·Moderate
Open
Ellipse latus rectum and foci
Let each of the two ellipses $E_1 : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ , $(a > b)$ and $E_2 : \dfrac{x^2}{A^2} + \dfrac{y^2}{B^2} = 1$ , $(A < B)$ have eccentricity $\dfrac45$ . Let the lengths of the latus recta of $E_1$ and $E_2$ be $\ell_1$ and $\ell_2$ , respectively, such that $2\ell_1^2 = 9\ell_2$ . If the distance between the foci of $E_1$ is 8, then the distance between the foci of $E_2$ is
1. (1) $\dfrac{96}{5}$
2. (2) $\dfrac{32}{5}$
3. (3) $\dfrac{16}{5}$
4. (4) $\dfrac85$
MathsJEE Main 24 Jan 2026, Shift 2·Q2·MCQ·Moderate
Open
Ellipse latus rectum and eccentricity
Let the length of the latus rectum of an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ , $(a > b)$ , be 30. If its eccentricity is the maximum value of the function $f(t) = -\dfrac34 + 2t - t^2$ , then $(a^2 + b^2)$ is equal to
1. (1)516
2. (2)256
3. (3)496
4. (4)276
MathsJEE Main 24 Jan 2026, Shift 2·Q6·MCQ·Moderate
Open
Reflection of parabola in line
Let the image of parabola $x^2 = 4y$ , in the line $x - y = 1$ be $(y + a)^2 = b(x - c)$ , $a, b, c \in \mathbb{N}$ . Then $a + b + c$ is equal to
1. (1)12
2. (2)4
3. (3)6
4. (4)8
We corrected a printing error in the original paper
Source printed “ $(y + \alpha)^2 = b(x - c)$ ” (Greek $\alpha$ for the $a$ of “ $a, b, c \in \mathbb{N}$ ” and “ $a + b + c$ ”).
MathsJEE Main 24 Jan 2026, Shift 2·Q22·Numerical·Easy–moderate
Open
Locus and eccentricity of ellipse
Let $(h, k)$ lie on the circle $C : x^2 + y^2 = 4$ and the point $(2h + 1, 3k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\dfrac{5}{e^2}$ is equal to .
MathsJEE Main 28 Jan 2026, Shift 1·Q22·Numerical·Moderate
Open
Eccentricity and latus rectum
For some $\theta \in \left(0, \dfrac{\pi}{2}\right)$ , let the eccentricity and the length of the latus rectum of the hyperbola $x^2 - y^2\sec^2\theta = 8$ be $e_1$ and $\ell_1$ , respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2\sec^2\theta + y^2 = 6$ be $e_2$ and $\ell_2$ , respectively. If $e_1^2 = e_2^2\left(\sec^2\theta + 1\right)$ , then $\left(\dfrac{\ell_1\ell_2}{e_1e_2}\right)\tan^2\theta$ is equal to .
MathsJEE Main 28 Jan 2026, Shift 2·Q3·MCQ·Moderate
Open
Parabola parametric chord and centroid
Let $A$ be the focus of the parabola $y^2 = 8x$ . Let the line $y = mx + c$ intersect the parabola at two distinct points $B$ and $C$ . If the centroid of the triangle $ABC$ is $\left(\dfrac73, \dfrac43\right)$ , then $(BC)^2$ is equal to :
1. (1)41
2. (2)80
3. (3)89
4. (4)32
MathsJEE Main 28 Jan 2026, Shift 2·Q9·MCQ·Easy–moderate
Open
Ellipse latus rectum
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 :
1. (1) $\dfrac{16}{\sqrt3}$
2. (2)6
3. (3) $4\sqrt3$
4. (4) $6\sqrt3$
MathsJEE Main 28 Jan 2026, Shift 2·Q10·MCQ·Moderate
Open
Confocal ellipse and hyperbola
Let the ellipse $E : \dfrac{x^2}{144} + \dfrac{y^2}{169} = 1$ and the hyperbola $H : \dfrac{x^2}{16} - \dfrac{y^2}{\lambda^2} = -1$ have the same foci. If $e$ and $L$ respectively denote the eccentricity and the length of the latus rectum of $H$ , then the value of $24(e + L)$ is :
1. (1)296
2. (2)126
3. (3)148
4. (4)67
MathsJEE Main 2 Apr 2026, Shift 1·Q10·MCQ·Easy–moderate
Open
Ellipse latus rectum
Let an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ , $a < b$ , pass through the point $(4, 3)$ and have eccentricity $\dfrac{\sqrt5}{3}$ . Then the length of its latus rectum is:
1. (1) $\dfrac{4\sqrt5}{3}$
2. (2) $2\sqrt5$
3. (3) $\dfrac{7\sqrt5}{3}$
4. (4) $\dfrac{8\sqrt5}{3}$
MathsJEE Main 2 Apr 2026, Shift 2·Q11·MCQ·Moderate
Open
Chord of rectangular hyperbola
Let $O$ be the origin, and $P$ and $Q$ be two points on the rectangular hyperbola $xy = 12$ such that the mid point of the line segment $PQ$ is $\left(\dfrac12, -\dfrac12\right)$ . Then the area of the triangle $OPQ$ equals:
1. (1) $\dfrac32$
2. (2) $\dfrac52$
3. (3) $\dfrac72$
4. (4) $\dfrac92$
MathsJEE Main 2 Apr 2026, Shift 2·Q23·Numerical·Moderate
Open
Locus as an ellipse
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 $C$ is $e$ , then $72e^2$ is equal to .
MathsJEE Main 4 Apr 2026, Shift 1·Q22·Numerical·Moderate
Open
Latus rectum of parabola and ellipse
Consider the parabola $P : y^2 = 4kx$ and the ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ . Let the line segment joining the points of intersection of $P$ and $E$ , be their latus rectums. If the eccentricity of $E$ is $e$ , then $e^2 + 2\sqrt2$ is equal to .
MathsJEE Main 4 Apr 2026, Shift 2·Q10·MCQ·Moderate
Open
Parametric point on ellipse
Let $P(3\cos\alpha, 2\sin\alpha)$ , $\alpha \neq 0$ , be a point on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ , $Q$ be a point on the circle $x^2 + y^2 - 14x - 14y + 82 = 0$ and $R$ be a point on the line $x + y = 5$ such that the centroid of the triangle $PQR$ is $\left(2 + \cos\alpha, 3 + \dfrac23\sin\alpha\right)$ . Then the sum of the ordinates of all possible points $R$ is:
1. (1)6
2. (2)2
3. (3)4
4. (4)8
MathsJEE Main 4 Apr 2026, Shift 2·Q11·MCQ·Moderate
Open
Hyperbola foci and directrices
Let $H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ be a hyperbola such that the distance between its foci is 6 and the distance between its directrices is $\dfrac83$ . If the line $x = \alpha$ intersects the hyperbola $H$ at the points $A$ and $B$ such that the area of the triangle $AOB$ is $4\sqrt{15}$ , where $O$ is the origin, then $\alpha^2$ equals
1. (1)12
2. (2)16
3. (3)24
4. (4)25
MathsJEE Main 4 Apr 2026, Shift 2·Q24·Numerical·Tough
Open
Locus of centroid on parabola
Let $A$ , $B$ and $C$ be the vertices of a variable right angled triangle inscribed in the parabola $y^2 = 16x$ . Let the vertex $B$ containing the right angle be $(4, 8)$ and the locus of the centroid of $\triangle ABC$ be a conic $C_0$ . Then three times the length of latus rectum of $C_0$ is .
MathsJEE Main 5 Apr 2026, Shift 1·Q9·MCQ·Easy–moderate
Open
Ellipse focus and eccentricity
Let a focus of the ellipse $E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ be $S(4, 0)$ and its eccentricity be $\dfrac45$ . If the point $P(3, \alpha)$ lies on $E$ and $O$ is the origin, then the area of $\Delta POS$ is equal to:
1. (1) $12/5$
2. (2) $14/5$
3. (3) $24/5$
4. (4) $48/5$
MathsJEE Main 5 Apr 2026, Shift 2·Q12·MCQ·Moderate
Open
Parabola focal chord
Let the directrix of the parabola $P : y^2 = 8x$ , cut $x$ -axis at the point $A$ . Let $B(\alpha, \beta)$ , $\alpha > 1$ , be a point on $P$ such that the slope of $AB$ is $3/5$ . If $BC$ is a focal chord of $P$ , then six times the area of $\Delta ABC$ is:
1. (1)80
2. (2)160
3. (3)174
4. (4)192
MathsJEE Main 5 Apr 2026, Shift 2·Q13·MCQ·Easy–moderate
Open
Hyperbola latus rectum
Let the eccentricity $e$ of a hyperbola satisfy the equation $6e^2 - 11e + 3 = 0$ . If the foci of the hyperbola are $(3, 5)$ $(3, -4)$ , then the length of its latus rectum is:
1. (1) $\dfrac{11}{3}$
2. (2) $\dfrac{17}{3}$
3. (3) $\dfrac{15}{2}$
4. (4) $\dfrac{17}{2}$
MathsJEE Main 6 Apr 2026, Shift 1·Q3·MCQ·Moderate
Open
Eccentricity of ellipse and hyperbola
Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2 - ax + 2 = 0$ . Let the sets $\{a \in \mathbb{R} : e_1$ and $e_2$ are the eccentricities of hyperbolas $\} = (\alpha, \beta)$ , and $\{a \in \mathbb{R} : e_1$ and $e_2$ are the eccentricities of an ellipse and a hyperbola, respectively $\} = (\gamma, \infty)$ . The $\alpha^2 + \beta^2 + \gamma^2$ is equal to:
1. (1)18
2. (2)22
3. (3)26
4. (4)34
MathsJEE Main 6 Apr 2026, Shift 1·Q11·MCQ·Moderate
Open
Hyperbola eccentricity and latus rectum
If the eccentricity $e$ of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ , passing through $\left(6, 4\sqrt3\right)$ , satisfies $15(e^2 + 1) = 34e$ , then the length of the latus rectum of the hyperbola $\dfrac{x^2}{b^2} - \dfrac{y^2}{2(a^2 + 1)} = 1$ is:
1. (1)10
2. (2)20
3. (3)25
4. (4)30
MathsJEE Main 6 Apr 2026, Shift 1·Q12·MCQ·Moderate
Open
Parametric chord of a parabola
Let chord $PQ$ of length $3\sqrt{13}$ of the parabola $y^2 = 12x$ be such that the ordinates of points $P$ and $Q$ are in the ratio $1:2$ . If the chord $PQ$ subtends an angle $\alpha$ at the focus of the parabola, then $\sin\alpha$ is equal to:
1. (1) $\dfrac35$
2. (2) $\dfrac45$
3. (3) $\dfrac{5}{13}$
4. (4) $\dfrac{12}{13}$
MathsJEE Main 6 Apr 2026, Shift 2·Q11·MCQ·Moderate
Open
Ellipse and hyperbola parameters
The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\dfrac{\sqrt3}{2}$ and its directrices are $x = \pm\dfrac{4\sqrt6}{3}$ . Let $H : \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of $E$ , and whose length of latus rectum is equal to the length of minor axis of $E$ . Then the distance between the foci of $H$ is:
1. (1) $\dfrac{4\sqrt2}{\sqrt7}$
2. (2) $\dfrac{4\sqrt2}{7}$
3. (3) $\dfrac{4}{\sqrt7}$
4. (4) $\dfrac{8}{7}$
MathsJEE Main 6 Apr 2026, Shift 2·Q12·MCQ·Moderate
Open
Locus of chord midpoint ellipse
Let $x = 9$ be a directrix of an ellipse $E$ , whose centre is at the origin and eccentricity is $\dfrac13$ . Let $P(\alpha, 0)$ , $\alpha > 0$ , be a focus of $E$ and $AB$ be a chord passing through $P$ . Then the locus of the mid point of $AB$ is:
1. (1) $9y^2 = 8x(1 - x)$
2. (2) $3y^2 = 4x(1 - x)$
3. (3) $9y^2 = 8x(x - 1)$
4. (4) $3y^2 = 4x(x - 1)$
MathsJEE Main 8 Apr 2026, Shift 2·Q11·MCQ·Moderate
Open
Parabola chord locus
Let $O$ be the vertex of the parabola $y^2 = 4x$ and its chords $OP$ and $OQ$ are perpendicular to each other. If the locus of the mid-point of the line segment $PQ$ is a conic $C$ , then the length of its latus rectum is:
1. (1)1
2. (2)2
3. (3)4
4. (4)8
MathsJEE Main 8 Apr 2026, Shift 2·Q20·MCQ·Easy–moderate
Open
Ellipse with major axis condition
Let $\dfrac{x^2}{f(a^2 + 7a + 3)} + \dfrac{y^2}{f(3a + 15)} = 1$ represent an ellipse with major axis along $y$ -axis, where $f$ is a strictly decreasing positive function on $\mathbb{R}$ . If the set of all possible values of $a$ is $\mathbb{R} - [\alpha, \beta]$ , then $\alpha^2 + \beta^2$ is equal to:
1. (1)28
2. (2)40
3. (3)61
4. (4)24