Conic Sections · JEE Main previous year questions

2019 27 questions

1. Q232019 · 12 Apr 2019 · Shift 1

   Let P be the point of intersection of the common tangents to the parabola $y^2=12x$ and the hyperbola $8x^2-y^2=8$. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS' in a ratio :

   1. A.$14:13$
   2. B.$13:11$
   3. C.$5:4$
   4. D.$2:1$
   Show answer

   Answer: C

2. Q262019 · 12 Apr 2019 · Shift 1

   If the normal to the ellipse $3x^2+4y^2=12$ 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 equal to :

   1. A.$\frac{\sqrt{61}}{2}$
   2. B.$\frac{\sqrt{221}}{2}$
   3. C.$\frac{\sqrt{157}}{2}$
   4. D.$\frac{5\sqrt{5}}{2}$
   Show answer

   Answer: D

3. Q42019 · 12 Apr 2019 · Shift 2

   If the area (in sq. units) bounded by the parabola $y^2=4\lambda x$ and the line $y=\lambda x$, $\lambda >0$, is $\frac{1}{9}$ , then $\lambda$ is equal to :

   1. A.$4\sqrt{3}$
   2. B.$2\sqrt{6}$
   3. C.48
   4. D.24
   Show answer

   Answer: D

4. Q92019 · 12 Apr 2019 · Shift 2

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

   1. A.$(2,\sqrt{2})$
   2. B.$(2,2\sqrt{2})$
   3. C.$(\sqrt{2},2)$
   4. D.$(1,2\sqrt{2})$
   Show answer

   Answer: C

5. Q112019 · 10 Apr 2019 · Shift 1

   If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2\sqrt{3})$ is $5x=4\sqrt{5}$ and its eccentricity is e, then :

   1. A.$4e^4-24e^2+27=0$
   2. B.$4e^4-24e^2+35=0$
   3. C.$4e^4-12e^2-27=0$
   4. D.$4e^4+8e^2-35=0$
   Show answer

   Answer: B

6. Q302019 · 10 Apr 2019 · Shift 1

   If the line $x-2y=12$ is tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\left(3,-\frac{9}{2}\right)$ , then the length of the latus rectum of the ellipse is :

   1. A.5
   2. B.9
   3. C.$8\sqrt{3}$
   4. D.$12\sqrt{2}$
   Show answer

   Answer: B

7. Q102019 · 10 Apr 2019 · Shift 2

   If $5x+9=0$ is the directrix of the hyperbola $16x^2-9y^2=144$, then its corresponding focus is :

   1. A.$\left(\frac{5}{3},0\right)$
   2. B.(5, 0)
   3. C.(- 5, 0)
   4. D.$\left(-\frac{5}{3},0\right)$
   Show answer

   Answer: C

8. Q272019 · 10 Apr 2019 · Shift 2

   The tangent and normal to the ellipse $3x^2+5y^2=32$ 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 :

   1. A.$\frac{14}{3}$
   2. B.$\frac{16}{3}$
   3. C.$\frac{68}{15}$
   4. D.$\frac{34}{15}$
   Show answer

   Answer: C

9. Q212019 · 9 Apr 2019 · Shift 1

   If one end of a focal chord of the parabola, $y^2=16x$ is at (1, 4), then the length of this focal chord is :

   1. A.24
   2. B.20
   3. C.25
   4. D.22
   Show answer

   Answer: C

10. Q282019 · 9 Apr 2019 · Shift 1

    If the line $y=mx+7\sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$, then a value of m is :

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

    Answer: D

11. Q222019 · 9 Apr 2019 · Shift 2

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

    1. A.$\sqrt{2}+1$
    2. B.$\sqrt{2}-1$
    3. C.$2\sqrt{2}+1$
    4. D.$2\sqrt{2}-1$
    Show answer

    Answer: A

12. Q152019 · 8 Apr 2019 · Shift 2

    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 $(0,5\sqrt{3})$, then the length of its latus rectum is :

    1. A.5
    2. B.8
    3. C.10
    4. D.6
    Show answer

    Answer: A

13. Q232019 · 8 Apr 2019 · Shift 2

    If the eccentricity of the standard hyperbola passing through the point (4,6) is 2, then the equation of the tangent to the hyperbola at (4,6) is :

    1. A.2x – y – 2 = 0
    2. B.3x – 2y = 0
    3. C.2x – 3y + 10 = 0
    4. D.x – 2y + 8 = 0
    Show answer

    Answer: A

14. Q282019 · 8 Apr 2019 · Shift 2

    The tangent to the parabola $y^2=4x$ at the point where it intersects the circle $x^2+y^2=5$ in the first quadrant, passes through the point :

    1. A.$\left(-\frac{1}{4},\frac{1}{2}\right)$
    2. B.$\left(-\frac{1}{3},\frac{4}{3}\right)$
    3. C.$\left(\frac{3}{4},\frac{7}{4}\right)$
    4. D.$\left(\frac{1}{4},\frac{3}{4}\right)$
    Show answer

    Answer: C

15. Q12019 · 12 Jan 2019 · Shift 1

    The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is :

    1. A.36
    2. B.$20\sqrt{2}$
    3. C.$18\sqrt{3}$
    4. D.32
    Show answer

    Answer: D

16. Q72019 · 12 Jan 2019 · Shift 1

    If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0)$, then which one of the following points does not lie on this hyperbola?

    1. A.$(6,5\sqrt{2})$
    2. B.$(2\sqrt{6},5)$
    3. C.$(-6,2\sqrt{10})$
    4. D.$(4,\sqrt{15})$
    Show answer

    Answer: A

17. Q142019 · 12 Jan 2019 · Shift 1

    The area (in sq. units) of the region bounded by the parabola, $y=x^2+2$ and the lines, $y=x+1$, $x=0$ and $x=3$, is

    1. A.$\frac{15}{4}$
    2. B.$\frac{15}{2}$
    3. C.$\frac{21}{2}$
    4. D.$\frac{17}{4}$
    Show answer

    Answer: B

18. Q222019 · 12 Jan 2019 · Shift 1

    Let $P(4,-4)$ and $Q(9,6)$ be two points on the parabola, $y^2=4x$ and let X be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of $\Delta \mathrm{PXQ}$ is maximum. Then this maximum area (in sq. units) is :

    1. A.$\frac{625}{4}$
    2. B.$\frac{125}{4}$
    3. C.$\frac{75}{2}$
    4. D.$\frac{125}{2}$
    Show answer

    Answer: B

19. Q92019 · 12 Jan 2019 · Shift 2

    Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $\Delta {\mathrm{S}}^{\prime }\mathrm{BS}$ is a right angled triangle with right angle at B and area $(\Delta {\mathrm{S}}^{\prime }\mathrm{BS})=8sq.\mathrm{units}$, then the length of a latus rectum of the ellipse is :

    1. A.2
    2. B.$4\sqrt{2}$
    3. C.4
    4. D.$2\sqrt{2}$
    Show answer

    Answer: C

20. Q232019 · 12 Jan 2019 · Shift 2

    The equation of a tangent to the parabola, $x^2=8y$, which makes an angle $\theta$ with the positive directions of x-axis, is :

    1. A.$x=y\cot \theta -2\tan \theta$
    2. B.$y=x\tan \theta +2\cot \theta$
    3. C.$x=y\cot \theta +2\tan \theta$
    4. D.$y=x\tan \theta -2\cot \theta$
    Show answer

    Answer: C

21. Q92019 · 11 Jan 2019 · Shift 2

    The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^2+1$, the tangent to it at the point $(2,5)$ and the coordinate axes is :

    1. A.$\frac{8}{3}$
    2. B.$\frac{14}{3}$
    3. C.$\frac{187}{24}$
    4. D.$\frac{37}{24}$
    Show answer

    Answer: D

22. Q142019 · 11 Jan 2019 · Shift 2

    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 the length of its minor axis, then which one of the following points lies on it?

    1. A.$(4\sqrt{2},2\sqrt{3})$
    2. B.$(4\sqrt{3},2\sqrt{3})$
    3. C.$(4\sqrt{3},2\sqrt{2})$
    4. D.$(4\sqrt{2},2\sqrt{2})$
    Show answer

    Answer: C

23. Q152019 · 11 Jan 2019 · Shift 2

    If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is :

    1. A.$\frac{13}{6}$
    2. B.2
    3. C.$\frac{13}{12}$
    4. D.$\frac{13}{8}$
    Show answer

    Answer: C

24. Q222019 · 11 Jan 2019 · Shift 2

    If the area of the triangle whose one vertex is at the vertex of the parabola, $y^2+4(x-a^2)=0$ and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is :

    1. A.$5\sqrt{5}$
    2. B.$(10{)}^{2/3}$
    3. C.$5(2^{1/3})$
    4. D.5
    Show answer

    Answer: D

25. Q102019 · 9 Jan 2019 · Shift 1

    Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which of the following points does not lie on it?

    1. A.$(5,2\sqrt{6})$
    2. B.$(6,4\sqrt{2})$
    3. C.$(8,6)$
    4. D.$(4,-4)$
    Show answer

    Answer: C