Conic Sections · JEE Main previous year questions

2023 26 questions

1. Q22023 · 13 Apr 2023 · Shift 1

   Let $\mathrm{PQ}$ be a focal chord of the parabola $y^2=36x$ of length $100$, making an acute angle with the positive $x$-axis. Let the ordinate of $\mathrm{P}$ be positive and $\mathrm{M}$ be the point on the line segment $\mathrm{PQ}$ such that $\mathrm{PM}:\mathrm{MQ}=3:1$. Then which of the following points does NOT lie on the line passing through $\mathrm{M}$ and perpendicular to the line $\mathrm{PQ}$?

   1. A.$(6,29)$
   2. B.$(-3,43)$
   3. C.$(3,33)$
   4. D.$(-6,45)$
   Show answer

   Answer: B

2. Q282023 · 13 Apr 2023 · Shift 1· Numerical answer

   Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $\mathrm{P}(4,1)$ to the hyperbola $H:\frac{y^2}{25}-\frac{x^2}{16}=1$. If Q is the point from which the tangents drawn to H have slopes $|m_1|$ and $|m_2|$ and they make positive intercepts $\alpha$ and $\beta$ on the $x$-axis, then $\frac{(PQ{)}^2}{\alpha \beta }$ is equal to _________.

   Show answer

   Answer: 8

3. Q302023 · 13 Apr 2023 · Shift 2· Numerical answer

   The foci of a hyperbola are $(\pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2x+3y=6$, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x$ - and $y$-axes are $a$ and $b$ respectively, then $|6a|+|5b|$ is equal to ___________

   Show answer

   Answer: 12

4. Q92023 · 12 Apr 2023 · Shift 1

   Let $\mathrm{P}\left(\frac{2\sqrt{3}}{\sqrt{7}},\frac{6}{\sqrt{7}}\right)$, $\mathrm{Q}$, $\mathrm{R}$ and $\mathrm{S}$ be four points on the ellipse $9x^2+4y^2=36$. Let $\mathrm{PQ}$ and $\mathrm{RS}$ be mutually perpendicular and pass through the origin. If $\frac{1}{(\mathrm{PQ}{)}^2}+\frac{1}{(\mathrm{RS}{)}^2}=\frac{p}{q}$, where $p$ and $q$ are coprime, then $p+q$ is equal to :

   1. A.$143$
   2. B.$147$
   3. C.$137$
   4. D.$157$
   Show answer

   Answer: D

5. Q222023 · 11 Apr 2023 · Shift 1· Numerical answer

   Let ${\text{H}}_n:\frac{x^2}{1+n}-\frac{y^2}{3+n}=1,n\in \mathbb{N}$. Let k be the smallest even value of n such that the eccentricity of ${\text{H}}_k$ is a rational number. If $l$ is the length of the latus rectum of ${\text{H}}_k$, then $21l$ is equal to _________.

   Show answer

   Answer: 306

6. Q292023 · 11 Apr 2023 · Shift 2· Numerical answer

   Let the tangent to the parabola $y^2=12x$ at the point $(3,\alpha )$ be perpendicular to the line $2x+2y=3$. Then the square of distance of the point $(6,-4)$ from the normal to the hyperbola ${\alpha }^2{x}^2-9y^2=9{\alpha }^2$ at its point $(\alpha -1,\alpha +2)$ is equal to _________.

   Show answer

   Answer: 116

7. Q32023 · 10 Apr 2023 · Shift 1

   Let the ellipse $E:x^2+9y^2=9$ intersect the positive x and y-axes at the points A and B respectively. Let the major axis of E be a diameter of the circle C. Let the line passing through A and B meet the circle C at the point P. If the area of the triangle with vertices A, P and the origin O is $\frac{m}{n}$, where m and n are coprime, then $m-n$ is equal to :

   1. A.15
   2. B.16
   3. C.17
   4. D.18
   Show answer

   Answer: C

8. Q222023 · 10 Apr 2023 · Shift 1· Numerical answer

   Let $y=p(x)$ be the parabola passing through the points $(-1,0)$, $(0,1)$ and $(1,0)$. If the area of the region $\{(x,y):(x+1{)}^2+(y-1{)}^2\le 1,y\le p(x)\}$ is A, then $12(\pi -4A)$ is equal to _________.

   Show answer

   Answer: 16

9. Q82023 · 8 Apr 2023 · Shift 1

   Let $R$ be the focus of the parabola $y^2=20x$ and the line $y=mx+c$ intersect the parabola at two points $P$ and $Q$. Let the point $G(10,10)$ be the centroid of the triangle $PQR$. If $c-m=6$, then $(PQ{)}^2$ is :

   1. A.$317$
   2. B.$325$
   3. C.$346$
   4. D.$296$
   Show answer

   Answer: B

10. Q202023 · 8 Apr 2023 · Shift 2

    Let $\mathrm{A}(0,1),\mathrm{B}(1,1)$ and $\mathrm{C}(1,0)$ be the mid-points of the sides of a triangle with incentre at the point $\mathrm{D}$. If the focus of the parabola $y^2=4ax$ passing through $\mathrm{D}$ is $(\alpha +\beta \sqrt{2},0)$, where $\alpha$ and $\beta$ are rational numbers, then $\frac{\alpha }{{\beta }^2}$ is equal to :

    1. A.$\frac{9}{2}$
    2. B.$12$
    3. C.$6$
    4. D.$8$
    Show answer

    Answer: D

11. Q292023 · 8 Apr 2023 · Shift 2· Numerical answer

    The ordinates of the points $\mathrm{P}$ and $\mathrm{Q}$ on the parabola with focus $(3,0)$ and directrix $x=-3$ are in the ratio $3:1$. If $\mathrm{R}(\alpha ,\beta )$ is the point of intersection of the tangents to the parabola at $\mathrm{P}$ and $\mathrm{Q}$, then $\frac{{\beta }^2}{\alpha }$ is equal to ___________.

    Show answer

    Answer: 16

12. Q282023 · 6 Apr 2023 · Shift 1· Numerical answer

    Let the tangent to the curve $x^2+2x-4y+9=0$ at the point $\text{P}(1,3)$ on it meet the $y$-axis at A. Let the line passing through P and parallel to the line $x-3y=6$ meet the parabola $y^2=4x$ at B. If B lies on the line $2x-3y=8$, then $(\text{AB}{)}^2$ is equal to ________.

    Show answer

    Answer: 292

13. Q102023 · 6 Apr 2023 · Shift 2

    In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse $25\left({\beta }^2{x}^2+{\alpha }^2{y}^2\right)={\alpha }^2{\beta }^2$ is :

    1. A.$\frac{\sqrt{129}}{12}$
    2. B.$\frac{3\sqrt{15}}{12}$
    3. C.$\frac{\sqrt{119}}{12}$
    4. D.$\frac{\sqrt{117}}{12}$
    Show answer

    Answer: C

14. Q222023 · 6 Apr 2023 · Shift 2· Numerical answer

    Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is reciprocal to that of the hyperbola $2x^2-2y^2=1$. If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectum of the ellipse is ________.

    Show answer

    Answer: 2

15. Q182023 · 31 Jan 2023 · Shift 1

    If the maximum distance of normal to the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1,b<2$, from the origin is 1, then the eccentricity of the ellipse is :

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

    Answer: D

16. Q202023 · 31 Jan 2023 · Shift 1

    Let $y=f(x)$ represent a parabola with focus $(-\frac{1}{2},0)$ and directrix $y=-\frac{1}{2}$. Then $S=\left\{x\in \mathbb{R}:{\tan }^{-1}(\sqrt{f(x)})+{\sin }^{-1}(\sqrt{f(x)+1})=\frac{\pi }{2}\right\}:$

    1. A.is an empty set
    2. B.contains exactly one element
    3. C.contains exactly two elements
    4. D.is an infinite set
    Show answer

    Answer: C

17. Q122023 · 31 Jan 2023 · Shift 2

    Let $\mathrm{H}$ be the hyperbola, whose foci are $(1\pm \sqrt{2},0)$ and eccentricity is $\sqrt{2}$. Then the length of its latus rectum is :

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

    Answer: C

18. Q292023 · 31 Jan 2023 · Shift 2· Numerical answer

    Let S be the set of all $a\in \mathbb{N}$ such that the area of the triangle formed by the tangent at the point $\mathrm{P}(b,c),b,c\in \mathbb{N}$, on the parabola $y^2=2ax$ and the lines $x=b,y=0$ is $16{\text{ unit}}^2$, then ${\sum }_{a\in \mathrm{S}}a$ is equal to :

    Show answer

    Answer: 146

19. Q142023 · 30 Jan 2023 · Shift 1

    If $\mathrm{P}(h,k)$ be a point on the parabola $x=4y^2$, which is nearest to the point $\mathrm{Q}(0,33)$, then the distance of $\mathrm{P}$ from the directrix of the parabola $y^2=4(x+y)$ is equal to :

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

    Answer: C

20. Q182023 · 29 Jan 2023 · Shift 2

    If the tangent at a point P on the parabola $y^2=3x$ is parallel to the line $x+2y=1$ and the tangents at the points Q and R on the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ are perpendicular to the line $x-y=2$, then the area of the triangle PQR is :

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

    Answer: B

21. Q272023 · 25 Jan 2023 · Shift 1· Numerical answer

    If the area enclosed by the parabolas ${\mathrm{P}}_1:2y=5x^2$ and ${\mathrm{P}}_2:x^2-y+6=0$ is equal to the area enclosed by ${\mathrm{P}}_1$ and $y=\alpha x,\alpha >0$, then ${\alpha }^3$ is equal to ____.

    Show answer

    Answer: 600

22. Q292023 · 25 Jan 2023 · Shift 1· Numerical answer

    The vertices of a hyperbola $\mathrm{H}$ are $(\pm 6,0)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $\mathrm{N}$ be the normal to $\mathrm{H}$ at a point in the first quadrant and parallel to the line $\sqrt{2}x+y=2\sqrt{2}$. If $\mathrm{d}$ is the length of the line segment of $\mathrm{N}$ between $\mathrm{H}$ and the y-axis then ${\mathrm{d}}^2$ is equal to ____.

    Show answer

    Answer: 216

23. Q12023 · 25 Jan 2023 · Shift 2

    The equations of two sides of a variable triangle are $x=0$ and $y=3$, and its third side is a tangent to the parabola $y^2=6x$. The locus of its circumcentre is :

    1. A.$4y^2-18y-3x-18=0$
    2. B.$4y^2+18y+3x+18=0$
    3. C.$4y^2-18y+3x+18=0$
    4. D.$4y^2-18y-3x+18=0$
    Show answer

    Answer: C

24. Q82023 · 25 Jan 2023 · Shift 2

    Let T and C respectively be the transverse and conjugate axes of the hyperbola $16x^2-y^2+64x+4y+44=0$. Then the area of the region above the parabola $x^2=y+4$, below the transverse axis T and on the right of the conjugate axis C is :

    1. A.$4\sqrt{6}-\frac{28}{3}$
    2. B.$4\sqrt{6}-\frac{44}{3}$
    3. C.$4\sqrt{6}+\frac{28}{3}$
    4. D.$4\sqrt{6}+\frac{44}{3}$
    Show answer

    Answer: C

25. Q142023 · 24 Jan 2023 · Shift 1

    Let a tangent to the curve $y^2=24x$ meet the curve $xy=2$ at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :

    1. A.length of latus rectum 2
    2. B.directrix $4x=-3$
    3. C.directrix $4x=3$
    4. D.length of latus rectum $\frac{3}{2}$
    Show answer

    Answer: C