Conic Sections · JEE Main previous year questions

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2022 32 questions

Q72022 · 29 Jul 2022 · Shift 1
Let a line $L$ pass through the point of intersection of the lines $b x + 10 y - 8 = 0$ and $2 x - 3 y = 0$ , $b \in R - {\frac{4}{3}}$ . If the line $L$ also passes through the point $(1, 1)$ and touches the circle $17 (x^{2} + y^{2}) = 16$ , then the eccentricity of the ellipse $\frac{x ^{2}}{5} + \frac{y ^{2}}{b ^{2}} = 1$ is :
1. A. $\frac{2}{5}$
2. B. $\frac{3}{5}$
3. C. $\frac{1}{5}$
4. D. $\frac{2}{5}$
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Answer: B
Q122022 · 29 Jul 2022 · Shift 1
Let the focal chord of the parabola $P : y^{2} = 4 x$ along the line $L : y = m x + c$ , $m > 0$ meet the parabola at the points $M$ and $N$ . Let the line $L$ be a tangent to the hyperbola $H : x^{2} - y^{2} = 4$ . If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive x-axis, then the area of the quadrilateral $OMFN$ is :
1. A. $26$
2. B. $214$
3. C. $46$
4. D. $414$
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Answer: B
Q172022 · 28 Jul 2022 · Shift 1
If the tangents drawn at the points $P$ and $Q$ on the parabola $y^{2} = 2 x - 3$ intersect at the point $R (0, 1)$ , then the orthocentre of the triangle $P QR$ is :
1. A. $(0, 1)$
2. B. $(2, - 1)$
3. C. $(6, 3)$
4. D. $(2, 1)$
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Answer: B
Q302022 · 28 Jul 2022 · Shift 1· Numerical answer
For the hyperbola $H : x^{2} - y^{2} = 1$ and the ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $a > b > 0$ , let the (1) eccentricity of $E$ be reciprocal of the eccentricity of $H$ , and (2) the line $y = \frac{5}{2} x + K$ be a common tangent of $E$ and $H$ . Then $4 (a^{2} + b^{2})$ is equal to ____.
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Answer: 3
Q122022 · 28 Jul 2022 · Shift 2
Let the hyperbola $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ pass through the point $(22, - 22)$ . A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is e times the length of the latus rectum of H, where e is the eccentricity of H, then which of the following points lies on the parabola?
1. A. $(23, 32)$
2. B. $(33, - 62)$
3. C. $(3, - 6)$
4. D. $(36, 62)$
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Answer: B
Q262022 · 28 Jul 2022 · Shift 2· Numerical answer
Let the tangents at the points P and Q on the ellipse $\frac{x ^{2}}{2} + \frac{y ^{2}}{4} = 1$ meet at the point $R (2, 22 - 2)$ . If S is the focus of the ellipse on its negative major axis, then $SP^{2} + SQ^{2}$ is equal to _________.
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Answer: 13
Q232022 · 27 Jul 2022 · Shift 1· Numerical answer
An ellipse $E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ passes through the vertices of the hyperbola $H : \frac{x ^{2}}{49} - \frac{y ^{2}}{64} = - 1$ . Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$ , respectively. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$ . If $l$ is the length of the latus rectum of the ellipse $E$ , then the value of $113 l$ is equal to _________.
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Answer: 1552
Q272022 · 27 Jul 2022 · Shift 1· Numerical answer
If the length of the latus rectum of the ellipse $x^{2} + 4 y^{2} + 2 x + 8 y - λ = 0$ is 4 , and $l$ is the length of its major axis, then $λ + l$ is equal to _________.
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Answer: 75
Q152022 · 27 Jul 2022 · Shift 2
If the length of the latus rectum of a parabola, whose focus is $(a, a)$ and the tangent at its vertex is $x + y = a$ , is $16$ , then $∣ a ∣$ is equal to :
1. A. $22$
2. B. $23$
3. C. $42$
4. D. $4$
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Answer: C
Q172022 · 26 Jul 2022 · Shift 1
Let the tangent drawn to the parabola $y^{2} = 24 x$ at the point $(α, β)$ is perpendicular to the line $2 x + 2 y = 5$ . Then the normal to the hyperbola $\frac{x ^{2}}{α ^{2}} - \frac{y ^{2}}{β ^{2}} = 1$ at the point $(α + 4, β + 4)$ does NOT pass through the point :
1. A. $(25, 10)$
2. B. $(20, 12)$
3. C. $(30, 8)$
4. D. $(15, 13)$
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Answer: D
Q92022 · 25 Jul 2022 · Shift 2
If the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ meets the line $\frac{x}{7} + \frac{y}{2 6} = 1$ on the $x$ -axis and the line $\frac{x}{7} - \frac{y}{2 6} = 1$ on the $y$ -axis, then the eccentricity of the ellipse is :
1. A. $\frac{5}{7}$
2. B. $\frac{2 6}{7}$
3. C. $\frac{3}{7}$
4. D. $\frac{2 5}{7}$
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Answer: A
Q102022 · 25 Jul 2022 · Shift 2
Let the foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{7} = 1$ and the hyperbola $\frac{x ^{2}}{144} - \frac{y ^{2}}{α} = \frac{1}{25}$ coincide. Then the length of the latus rectum of the hyperbola is :
1. A. $\frac{32}{9}$
2. B. $\frac{18}{5}$
3. C. $\frac{27}{4}$
4. D. $\frac{27}{10}$
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Answer: D
Q182022 · 25 Jul 2022 · Shift 2
The tangents at the points $A (1, 3)$ and $B (1, - 1)$ on the parabola $y^{2} - 2 x - 2 y = 1$ meet at the point $P$ . Then the area (in $unit^{2}$ ) of the triangle $P A B$ is :
1. A.4
2. B.6
3. C.7
4. D.8
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Answer: D
Q102022 · 30 Jun 2022 · Shift 1
Let the eccentricity of the ellipse $x^{2} + a^{2} y^{2} = 25 a^{2}$ be b times the eccentricity of the hyperbola $x^{2} - a^{2} y^{2} = 5$ , where a is the minimum distance between the curves $y = e^{x}$ and $y = lo g_{e} x$ . Then $a^{2} + \frac{1}{b ^{2}}$ is equal to :
1. A. $\frac{3}{2}$
2. B. $\frac{5}{2}$
3. C.3
4. D.5
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Answer: D
Q262022 · 30 Jun 2022 · Shift 1· Numerical answer
Let PQ be a focal chord of length 6.25 units of the parabola $y^{2} = 4 x$ . If O is the vertex of the parabola, then 10 times the area (in sq. units) of $Δ POQ$ is equal to ____ .
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Answer: 25
Q132022 · 28 Jun 2022 · Shift 2
Let $a > 0, b > 0$ . Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . Let $e^{'}$ and $l^{'}$ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e^{2} = \frac{11}{14} l$ and $(e^{'})^{2} = \frac{11}{8} l^{'}$ , then the value of $77 a + 44 b$ is equal to :
1. A. $100$
2. B. $110$
3. C. $120$
4. D. $130$
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Answer: D
Q152022 · 28 Jun 2022 · Shift 2
If vertex of a parabola is $(2, - 1)$ and the equation of its directrix is $4 x - 3 y = 21$ , then the length of its latus rectum is :
1. A. $2$
2. B. $8$
3. C. $12$
4. D. $16$
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Answer: B
Q112022 · 27 Jun 2022 · Shift 1
Let the eccentricity of an ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $a > b$ , be $\frac{1}{4}$ . If this ellipse passes through the point $(- 4 \frac{2}{5}, 3)$ , then $a^{2} + b^{2}$ is equal to :
1. A.29
2. B.31
3. C.32
4. D.34
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Answer: B
Q72022 · 27 Jun 2022 · Shift 2
If the equation of the parabola, whose vertex is at $(5, 4)$ and the directrix is $3 x + y - 29 = 0$ , is $x^{2} + a y^{2} + b x y + c x + d y + k = 0$ , then $a + b + c + d + k$ is equal to :
1. A. $575$
2. B. $- 575$
3. C. $576$
4. D. $- 576$
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Answer: D
Q162022 · 26 Jun 2022 · Shift 1
Let the normal at the point $P$ on the parabola $y^{2} = 6 x$ pass through the point $(5, - 8)$ . If the tangent at $P$ to the parabola intersects its directrix at the point $Q$ , then the ordinate of the point $Q$ is :
1. A. $- 3$
2. B. $- \frac{9}{4}$
3. C. $- \frac{5}{2}$
4. D. $- 2$
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Answer: B
Q292022 · 26 Jun 2022 · Shift 1· Numerical answer
Let the common tangents to the curves $4 (x^{2} + y^{2}) = 9$ and $y^{2} = 4 x$ intersect at the point $Q$ . Let an ellipse, centered at the origin $O$ , has lengths of semi-minor and semi-major axes equal to $O Q$ and 6, respectively. If $e$ and $l$ respectively denote the eccentricity and the length of the latus rectum of this ellipse, then $\frac{l}{e ^{2}}$ is equal to ____.
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Answer: 4
Q102022 · 26 Jun 2022 · Shift 2
The locus of the mid point of the line segment joining the point $(4, 3)$ and the points on the ellipse $x^{2} + 2 y^{2} = 4$ is an ellipse with eccentricity :
1. A. $\frac{3}{2}$
2. B. $\frac{1}{2 2}$
3. C. $\frac{1}{2}$
4. D. $\frac{1}{2}$
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Answer: C
Q172022 · 26 Jun 2022 · Shift 2
The normal to the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{9} = 1$ at the point $(8, 33)$ on it passes through the point :
1. A. $(15, - 23)$
2. B. $(9, 23)$
3. C. $(- 1, 93)$
4. D. $(- 1, 63)$
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Answer: C
Q302022 · 26 Jun 2022 · Shift 2· Numerical answer
Let a line $L_{1}$ be tangent to the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{4} = 1$ and let $L_{2}$ be the line passing through the origin and perpendicular to $L_{1}$ . If the locus of the point of intersection of $L_{1}$ and $L_{2}$ is $(x^{2} + y^{2})^{2} = α x^{2} + β y^{2}$ , then $α + β$ is equal to ________.
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Answer: 12
Q62022 · 25 Jun 2022 · Shift 2
The area of the region enclosed between the parabolas $y^{2} = 2 x - 1$ and $y^{2} = 4 x - 3$ is
1. A. $\frac{1}{3}$
2. B. $\frac{1}{6}$
3. C. $\frac{2}{3}$
4. D. $\frac{3}{4}$
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Answer: A