Conic Sections · JEE Main previous year questions

2025 26 questions

1. Q42025 · 4 Apr 2025 · Shift 1

   The length of the latus-rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\frac{4}{5}$, is

   1. A.$\frac{50}{3}$
   2. B.$\frac{18}{5}$
   3. C.$\frac{6}{5}$
   4. D.$\frac{10}{3}$
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   Answer: B

2. Q42025 · 4 Apr 2025 · Shift 2

   Let the sum of the focal distances of the point $\mathrm{P}(4,3)$ on the hyperbola $\mathrm{H}:\frac{x^2}{{\mathrm{a}}^2}-\frac{y^2}{{\mathrm{b}}^2}=1$ be $8\sqrt{\frac{5}{3}}$. If for H , the length of the latus rectum is $l$ and the product of the focal distances of the point P is m , then $9l^2+6\mathrm{m}$ is equal to :

   1. A.187
   2. B.184
   3. C.186
   4. D.185
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   Answer: D

3. Q92025 · 4 Apr 2025 · Shift 2

   Let for two distinct values of p the lines $y=x+p$ touch the ellipse $\mathrm{E}:\frac{x^2}{4^2}+\frac{y^2}{3^2}=1$ 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 $ABCD$ is equal to :

   1. A.48
   2. B.20
   3. C.24
   4. D.36
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   Answer: C

4. Q102025 · 4 Apr 2025 · Shift 2

   The centre of a circle C is at the centre of the ellipse $\mathrm{E}:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$. Let C pass through the foci $F_1$ and $F_2$ of E such that the circle $C$ and the ellipse $E$ intersect at four points. Let P be one of these four points. If the area of the triangle ${\mathrm{PF}}_1{\mathrm{F}}_2$ is 30 and the length of the major axis of $E$ is 17 , then the distance between the foci of $E$ is :

   1. A.12
   2. B.26
   3. C.13
   4. D.$\frac{13}{2}$
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   Answer: C

5. Q112025 · 4 Apr 2025 · Shift 2

   A line passing through the point $\mathrm{A}(-2,0)$, touches the parabola $\mathrm{P}:y^2=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $AB$, parabola $P$ and the $x$-axis, is :

   1. A.3
   2. B.$\frac{7}{3}$
   3. C.$\frac{8}{3}$
   4. D.2
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   Answer: C

6. Q162025 · 4 Apr 2025 · Shift 2

   The axis of a parabola is the line $y=x$ and its vertex and focus are in the first quadrant at distances $\sqrt{2}$ and $2\sqrt{2}$ units from the origin, respectively. If the point $(1,k)$ lies on the parabola, then a possible value of $k$ is :

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

7. Q192025 · 4 Apr 2025 · Shift 2

   Let the values of p , for which the shortest distance between the lines $\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and $\vec{r}=(p\hat{i}+2\hat{j}+\hat{k})+\lambda (2\hat{i}+3\hat{j}+4\hat{k})$ is $\frac{1}{\sqrt{6}}$, be a, b, $(a<b)$. Then the length of the latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is :

   1. A.$\frac{3}{2}$
   2. B.9
   3. C.18
   4. D.$\frac{2}{3}$
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   Answer: D

8. Q122025 · 3 Apr 2025 · Shift 1

   A line passing through the point $P(\sqrt{5},\sqrt{5})$ intersects the ellipse $\frac{x^2}{36}+\frac{y^2}{25}=1$ at $A$ and $B$ such that $(PA)\cdot (PB)$ is maximum. Then $5\left(P{A}^2+P{B}^2\right)$ is equal to :

   1. A.290
   2. B.377
   3. C.338
   4. D.218
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   Answer: C

9. Q232025 · 3 Apr 2025 · Shift 1· Numerical answer

   Let the product of the focal distances of the point $\mathbf{P}(4,2\sqrt{3})$ on the hyperbola $\mathrm{H}:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be 32 . Let the length of the conjugate axis of H be $p$ and the length of its latus rectum be $q$. Then $p^2+q^2$ is equal to____

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   Answer: 120

10. Q52025 · 3 Apr 2025 · Shift 2

    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 $PQR$ be a variable triangle, whose vertex $P$ is on the circle $C$ and the side $QR$ of length $2a$ is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle $PQR$ is :

    1. A.$8(3+\sqrt{2})$
    2. B.$8(2+\sqrt{3})$
    3. C.$6(3+\sqrt{2})$
    4. D.$6(2+\sqrt{3})$
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    Answer: B

11. Q222025 · 3 Apr 2025 · Shift 2· Numerical answer

    If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3x^2-y^2-\alpha x+\beta y+\gamma =0$, then $\alpha +\beta +\gamma$ is equal to_________.

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    Answer: 141

12. Q52025 · 2 Apr 2025 · Shift 1

    If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and $P$ be a point on the ellipse, then $\min (SP\cdot S^{\prime }P)+\max (SP\cdot S^{\prime }P)$ is equal to :

    1. A.$3(6+\sqrt{2})$
    2. B.$3(1+\sqrt{2})$
    3. C.27
    4. D.9
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    Answer: C

13. Q62025 · 2 Apr 2025 · Shift 1

    Let the focal chord $PQ$ of the parabola $y^2=4x$ make an angle of $60^{\circ }$ with the positive $x$ axis, where $P$ lies in the first quadrant. If the circle, whose one diameter is $PS$, $S$ being the focus of the parabola, touches the $y$-axis at the point $(0,\alpha )$, then $5{\alpha }^2$ is equal to:

    1. A.15
    2. B.25
    3. C.20
    4. D.30
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    Answer: A

14. Q102025 · 2 Apr 2025 · Shift 1

    Let one focus of the hyperbola $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be at $(\sqrt{10},0)$ and the corresponding directrix be $x=\frac{9}{\sqrt{10}}$. If $e$ and $l$ respectively are the eccentricity and the length of the latus rectum of $H$, then $9(e^2+l)$ is equal to :

    1. A.12
    2. B.14
    3. C.15
    4. D.16
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    Answer: D

15. Q42025 · 2 Apr 2025 · Shift 2

    Let the point P of the focal chord PQ of the parabola $y^2=16x$ be $(1,-4)$. If the focus of the parabola divides the chord $PQ$ in the ratio $m:n,\gcd (m,n)=1$, then $m^2+n^2$ is equal to :

    1. A.17
    2. B.37
    3. C.10
    4. D.26
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    Answer: A

16. Q162025 · 2 Apr 2025 · Shift 2

    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 :

    1. A.$\frac{3}{\sqrt{19}}$
    2. B.$\frac{\sqrt{3}}{16}$
    3. C.$\frac{4}{\sqrt{17}}$
    4. D.$\frac{\sqrt{5}}{7}$
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    Answer: C

17. Q112025 · 29 Jan 2025 · Shift 2

    If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, whose mid point is $\left(\frac{5}{2},\frac{1}{2}\right)$. then $\alpha +\beta$ is equal to :

    1. A.$37$
    2. B.$46$
    3. C.$72$
    4. D.$58$
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    Answer: D

18. Q222025 · 29 Jan 2025 · Shift 2· Numerical answer

    Let $y^2=12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(SP)(SQ)=\frac{147}{4}$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2+64y^2-\alpha x-64\sqrt{3}y=\beta$, then $\beta -\alpha$ is equal to ____.

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    Answer: 1328

19. Q122025 · 28 Jan 2025 · Shift 1

    Let ABCD be a trapezium whose vertices lie on the parabola $y^2=4x$. Let the sides AD and BC of the trapezium be parallel to $y$-axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point $(1,0)$, then the area of $ABCD$ is

    1. A.$\frac{75}{8}$
    2. B.$\frac{125}{8}$
    3. C.$\frac{25}{2}$
    4. D.$\frac{75}{4}$
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    Answer: D

20. Q242025 · 28 Jan 2025 · Shift 1· Numerical answer

    Let ${\mathrm{E}}_1:\frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses ${\mathrm{E}}_i$ 's are constructed such that their centres and eccentricities are same as that of ${\mathrm{E}}_1$, and the length of minor axis of ${\mathrm{E}}_i$ is the length of major axis of ${\mathrm{E}}_{i+1}(i\ge 1)$. If $A_i$ is the area of the ellipse $E_i$, then $\frac{5}{\pi }\left({\sum }_{i=1}^{\infty }{A}_i\right)$, is equal to ____.

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    Answer: 54

21. Q82025 · 24 Jan 2025 · Shift 1

    Let the product of the focal distances of the point $\left(\sqrt{3},\frac{1}{2}\right)$ 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 is

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

22. Q122025 · 24 Jan 2025 · Shift 2

    If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2},3\right)$ and the directrix $x+2y=0$ is $\alpha x^2+\beta y^2-\gamma xy-30x-60y+225=0$, then $\alpha +\beta +\gamma$ is equal to :

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

23. Q142025 · 24 Jan 2025 · Shift 2

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

    1. A.$5x+16y=31$
    2. B.$48x+25y=169$
    3. C.$4x+122y=134$
    4. D.$25x+101y=176$
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    Answer: B

24. Q252025 · 24 Jan 2025 · Shift 2· Numerical answer

    Let ${\mathrm{H}}_1:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and ${\mathrm{H}}_2:-\frac{x^2}{{\mathrm{A}}^2}+\frac{y^2}{{\mathrm{B}}^2}=1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1=\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________ .

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    Answer: 55

25. Q32025 · 23 Jan 2025 · Shift 2

    The length of the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$, whose mid-point is $\left(1,\frac{1}{2}\right)$, is :

    1. A.$\frac{2}{3}\sqrt{15}$
    2. B.$\frac{1}{3}\sqrt{15}$
    3. C.$\sqrt{15}$
    4. D.$\frac{5}{3}\sqrt{15}$
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    Answer: A