Conic Sections · JEE Main previous year questions

2024 26 questions

1. Q132024 · 9 Apr 2024 · Shift 1

   Let $f(x)=x^2+9$, $g(x)=\frac{x}{x-9}$ and $\mathrm{a}=f\circ g(10)$, $\mathrm{b}=g\circ f(3)$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{\mathrm{a}}+\frac{y^2}{\mathrm{b}}=1$, then $8e^2+l^2$ is equal to:

   1. A.6
   2. B.12
   3. C.8
   4. D.16
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   Answer: C

2. Q12024 · 9 Apr 2024 · Shift 2

   Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E:\frac{(x-1{)}^2}{100}+\frac{(y-1{)}^2}{75}=1$ and the eccentricity of the hyperbola $H$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $H$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3{\alpha }^2+2{\beta }^2$ is equal to

   1. A.$225$
   2. B.$237$
   3. C.$242$
   4. D.$205$
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   Answer: A

3. Q172024 · 9 Apr 2024 · Shift 2

   The area (in square units) of the region enclosed by the ellipse $x^2+3y^2=18$ in the first quadrant below the line $y=x$ is

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

4. Q222024 · 9 Apr 2024 · Shift 2· Numerical answer

   Let $A,B$ and $C$ be three points on the parabola $y^2=6x$ and let the line segment $AB$ 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 $A$ and $B$ on $L$. Then ${\left(\frac{AM\cdot BN}{CD}\right)}^2$ is equal to ____.

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   Answer: 36

5. Q72024 · 8 Apr 2024 · Shift 1

   Let $H:-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha ,6),\alpha >0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha ,6)$, then ${\alpha }^2+\beta$ is equal to

   1. A.$170$
   2. B.$171$
   3. C.$169$
   4. D.$172$
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   Answer: B

6. Q262024 · 8 Apr 2024 · Shift 2· Numerical answer

   Let $\mathrm{A}$ be the region enclosed by the parabola $y^2=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $\mathrm{A}$ is ____.

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   Answer: 128

7. Q262024 · 6 Apr 2024 · Shift 2· Numerical answer

   The length of the latus rectum and directrices of hyperbola with eccentricity $\mathrm{e}$ are 9 and $x=\pm \frac{4}{\sqrt{13}}$, respectively. Let the line $y-\sqrt{3}x+\sqrt{3}=0$ touch this hyperbola at $(x_0,y_0)$. If $\mathrm{m}$ is the product of the focal distances of the point $(x_0,y_0)$, then $4{\mathrm{e}}^2+\mathrm{m}$ is equal to ____.

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   Answer: 61

8. Q232024 · 4 Apr 2024 · Shift 1· Numerical answer

   Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x,y)$ satisfying $A^2+xA+y\mathrm{I}=\mathrm{O}$ lie on a hyperbola, whose transverse axis is parallel to the $x$-axis, eccentricity is $e$ and the length of the latus rectum is $l$, then $e^4+l^4$ is equal to ____.

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   Answer: 25

9. Q282024 · 4 Apr 2024 · Shift 1· Numerical answer

   Let the length of the focal chord $\mathrm{PQ}$ of the parabola $y^2=12x$ be 15 units. If the distance of $\mathrm{PQ}$ from the origin is $\mathrm{p}$, then $10{\mathrm{p}}^2$ is equal to ____.

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   Answer: 72

10. Q12024 · 4 Apr 2024 · Shift 2

    Consider a hyperbola $\mathrm{H}$ having centre at the origin and foci on the $x$-axis. Let $C_1$ be the circle touching the hyperbola $\mathrm{H}$ and having the centre at the origin. Let $C_2$ be the circle touching the hyperbola $\mathrm{H}$ at its vertex and having the centre at one of its foci. If areas (in sq units) of $C_1$ and $C_2$ are $36\pi$ and $4\pi$, respectively, then the length (in units) of latus rectum of $\mathrm{H}$ is

    1. A.$\frac{28}{3}$
    2. B.$\frac{11}{3}$
    3. C.$\frac{14}{3}$
    4. D.$\frac{10}{3}$
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    Answer: A

11. Q132024 · 4 Apr 2024 · Shift 2

    Let $PQ$ be a chord of the parabola $y^2=12x$ and the midpoint of $PQ$ be at $(4,1)$. Then, which of the following point lies on the line passing through the points $\mathrm{P}$ and $\mathrm{Q}$ ?

    1. A.$(3,-3)$
    2. B.$\left(\frac{1}{2},-20\right)$
    3. C.$(2,-9)$
    4. D.$\left(\frac{3}{2},-16\right)$
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    Answer: B

12. Q132024 · 1 Feb 2024 · Shift 1

    For $0<\theta <\pi /2$, if the eccentricity of the hyperbola $x^2-y^2{\mathrm{cosec}}^2\theta =5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2{\mathrm{cosec}}^2\theta +y^2=5$, then the value of $\theta$ is :

    1. A.$\frac{\pi }{6}$
    2. B.$\frac{5\pi }{12}$
    3. C.$\frac{\pi }{3}$
    4. D.$\frac{\pi }{4}$
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    Answer: C

13. Q162024 · 1 Feb 2024 · Shift 1

    Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\mathrm{a}>\mathrm{b}$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is :

    1. A.3
    2. B.$\frac{7}{2}$
    3. C.$\frac{3}{2}$
    4. D.$\frac{5}{2}$
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    Answer: C

14. Q92024 · 1 Feb 2024 · Shift 2

    Let $\mathrm{P}$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through $\mathrm{P}$ and parallel to $y$-axis meet the circle $x^2+y^2=9$ at point $\mathrm{Q}$ such that $\mathrm{P}$ and $\mathrm{Q}$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $PQ$ such that $PR:RQ=4:3$ as $P$ moves on the ellipse, is :

    1. A.$\frac{13}{21}$
    2. B.$\frac{\sqrt{139}}{23}$
    3. C.$\frac{\sqrt{13}}{7}$
    4. D.$\frac{11}{19}$
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    Answer: C

15. Q212024 · 1 Feb 2024 · Shift 2· Numerical answer

    Three points $\mathrm{O}(0,0)$, $\mathrm{P}\left(\mathrm{a},{\mathrm{a}}^2\right)$, $\mathrm{Q}\left(-\mathrm{b},{\mathrm{b}}^2\right)$, $\mathrm{a}>0$, $\mathrm{b}>0$, are on the parabola $y=x^2$. Let ${\mathrm{S}}_1$ be the area of the region bounded by the line $\mathrm{PQ}$ and the parabola, and ${\mathrm{S}}_2$ be the area of the triangle $\mathrm{OPQ}$. If the minimum value of $\frac{{\mathrm{S}}_1}{{\mathrm{S}}_2}$ is $\frac{\mathrm{m}}{\mathrm{n}}$, $\gcd (\mathrm{m},\mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ________ .

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    Answer: 7

16. Q222024 · 1 Feb 2024 · Shift 2· Numerical answer

    The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2y^2=kx$ and $k{y}^2=2(y-x)$ is maximum, is equal to :

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    Answer: 8

17. Q202024 · 31 Jan 2024 · Shift 1

    If the foci of a hyperbola are same as that of the ellipse $\frac{x^2}{9}+\frac{y^2}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt{2},\frac{14}{3}\sqrt{\frac{2}{5}}\right)$ on the hyperbola, is equal to

    1. A.$14\sqrt{\frac{2}{5}}-\frac{4}{3}$
    2. B.$7\sqrt{\frac{2}{5}}+\frac{8}{3}$
    3. C.$7\sqrt{\frac{2}{5}}-\frac{8}{3}$
    4. D.$14\sqrt{\frac{2}{5}}-\frac{16}{3}$
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    Answer: C

18. Q272024 · 31 Jan 2024 · Shift 1· Numerical answer

    Let the foci and length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ be $(\pm 5,0)$ and $\sqrt{50}$, respectively. Then, the square of the eccentricity of the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2{b}^2}=1$ equals

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    Answer: 51

19. Q62024 · 31 Jan 2024 · Shift 2

    Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$, of eccentricity $\frac{1}{\sqrt{2}}$ pass through the focus of the parabola $P$. Then, the square of the length of the latus rectum of $E$, is

    1. A.$\frac{512}{25}$
    2. B.$\frac{656}{25}$
    3. C.$\frac{385}{8}$
    4. D.$\frac{347}{8}$
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    Answer: B

20. Q92024 · 31 Jan 2024 · Shift 2

    The area of the region enclosed by the parabolas $y=4x-x^2$ and $3y=(x-4{)}^2$ is equal to :

    1. A.$\frac{32}{9}$
    2. B.$\frac{14}{3}$
    3. C.4
    4. D.6
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    Answer: D

21. Q112024 · 30 Jan 2024 · Shift 1

    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 :

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

22. Q202024 · 30 Jan 2024 · Shift 1

    The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2x-8$, is :

    1. A.7
    2. B.8
    3. C.9
    4. D.6
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    Answer: C

23. Q262024 · 30 Jan 2024 · Shift 1· Numerical answer

    Let the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi }{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then $l^2+m^2+n^2$ is equal to _________.

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    Answer: 182

24. Q272024 · 29 Jan 2024 · Shift 2· Numerical answer

    Let $P(\alpha ,\beta )$ be a point on the parabola $y^2=4x$. If $P$ also lies on the chord of the parabola $x^2=8y$ whose mid point is $\left(1,\frac{5}{4}\right)$, then $(\alpha -28)(\beta -8)$ is equal to _________.

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    Answer: 192

25. Q82024 · 27 Jan 2024 · Shift 1

    The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1,\frac{2}{5}\right)$, is equal to :

    1. A.$\frac{\sqrt{1691}}{5}$
    2. B.$\frac{\sqrt{2009}}{5}$
    3. C.$\frac{\sqrt{1541}}{5}$
    4. D.$\frac{\sqrt{1741}}{5}$
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    Answer: A