Conic Sections · JEE Main previous year questions

2026 34 questions

1. Q82026 · 23 Jan 2026 · Shift 2

   Let PQ be a chord of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$, 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 $\sqrt{3}$, then the area of the triangle OPQ is

   1. A.$2\sqrt{3}$
   2. B.$\frac{11}{5}$
   3. C.$\frac{8\sqrt{3}}{5}$
   4. D.$\frac{9}{5}$
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   Answer: C

2. Q32026 · 22 Jan 2026 · Shift 1

   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. A.0.6
   2. B.0.7
   3. C.0.8
   4. D.0.5
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   Answer: C

3. Q82026 · 22 Jan 2026 · Shift 1

   If the chord joining the points ${\mathrm{P}}_1(x_1,y_1)$ and ${\mathrm{P}}_2(x_2,y_2)$ on the parabola $y^2=12x$ subtends a right angle at the vertex of the parabola, then $x_1{x}_2-y_1{y}_2$ is equal to

   1. A.280
   2. B.288
   3. C.292
   4. D.284
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   Answer: B

4. Q122026 · 21 Jan 2026 · Shift 1

   The area of the region, inside the ellipse $x^2+4y^2=4$ and outside the region bounded by the curves $y=|x|-1$ and $y=1-|x|$, is :

   1. A.$2\pi -1$
   2. B.$3(\pi -1)$
   3. C.$2(\pi -1)$
   4. D.$2\pi -\frac{1}{2}$
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   Answer: C

5. Q162026 · 21 Jan 2026 · Shift 1

   Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$. If the eccentricity of the hyperbola is 5 , then the length of its latus rectum is :

   1. A.$\frac{96}{\sqrt{5}}$
   2. B.$24\sqrt{5}$
   3. C.12
   4. D.16
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   Answer: A

6. Q182026 · 21 Jan 2026 · Shift 1

   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. A.$5x-4y+3=0$
   2. B.$x-2y+3=0$
   3. C.$4x-5y+6=0$
   4. D.$5x-y-3=0$
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   Answer: A

7. Q12026 · 21 Jan 2026 · Shift 2

   If the line $\alpha x+4y=\sqrt{7}$, where $\alpha \in \mathbf{R}$, 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. A.$\frac{1}{\sqrt{3}}-\frac{1}{2\sqrt{11}}$
   2. B.$\frac{1}{\sqrt{3}}-\frac{1}{2\sqrt{5}}$
   3. C.$\frac{1}{\sqrt{3}}+\frac{1}{2\sqrt{5}}$
   4. D.$\frac{1}{\sqrt{3}}+\frac{1}{2\sqrt{7}}$
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   Answer: D

8. Q32026 · 21 Jan 2026 · Shift 2

   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. A.$y^2-4x+8=0$
   2. B.$y^2-2x+8=0$
   3. C.$y^2-9x+6=0$
   4. D.$y^2-6x+4=0$
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   Answer: B

9. Q82026 · 21 Jan 2026 · Shift 2

   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. A.5
   2. B.7
   3. C.22
   4. D.16
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   Answer: B