Conic Sections · JEE Main previous year questions

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2020 22 questions

Q172020 · 6 Sept 2020 · Shift 1
Let $L_{1}$ be a tangent to the parabola $y^{2} = 4 (x + 1)$ and $L_{2}$ be a tangent to the parabola $y^{2} = 8 (x + 2)$ such that $L_{1}$ and $L_{2}$ intersect at right angles. Then $L_{1}$ and $L_{2}$ meet on the straight line :
1. A. $x + 3 = 0$
2. B. $x + 2 y = 0$
3. C. $x + 2 = 0$
4. D. $2 x + 1 = 0$
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Answer: A
Q192020 · 6 Sept 2020 · Shift 1
Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1$ from any of its foci?
1. A. $(- 1, 3)$
2. B. $(- 2, 3)$
3. C. $(- 1, 2)$
4. D. $(1, 2)$
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Answer: A
Q162020 · 6 Sept 2020 · Shift 2
If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity e of the ellipse satisfies :
1. A. $e^{4} + 2 e^{2} - 1 = 0$
2. B. $e^{4} + e^{2} - 1 = 0$
3. C. $e^{2} + 2 e - 1 = 0$
4. D. $e^{2} + e - 1 = 0$
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Answer: B
Q32020 · 3 Sept 2020 · Shift 1
A hyperbola having the transverse axis of length $2$ has the same foci as that of the ellipse $3 x^{2} + 4 y^{2} = 12$ , then this hyperbola does not pass through which of the following points?
1. A. $(1, - \frac{1}{2})$
2. B. $(\frac{3}{2}, \frac{1}{2})$
3. C. $(- \frac{3}{2}, 1)$
4. D. $(\frac{1}{2}, 0)$
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Answer: B
Q62020 · 3 Sept 2020 · Shift 1
Let P be a point on the parabola, $y^{2} = 12 x$ and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn through the mid-point M of PN, parallel to its axis which meets the parabola at Q. If the y-intercept of the line NQ is $\frac{4}{3}$ , then :
1. A. $MQ = \frac{1}{3}$
2. B. $PN = 4$
3. C. $PN = 3$
4. D. $MQ = \frac{1}{4}$
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Answer: D
Q62020 · 3 Sept 2020 · Shift 2
Let $e_{1}$ and $e_{2}$ be the eccentricities of the ellipse, $\frac{x ^{2}}{25} + \frac{y ^{2}}{b ^{2}} = 1 (b < 5)$ and the hyperbola, $\frac{x ^{2}}{16} - \frac{y ^{2}}{b ^{2}} = 1$ respectively satisfying $e_{1} e_{2} = 1$ . If $α$ and $β$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $(α, β)$ is equal to :
1. A. $(8, 10)$
2. B. $(8, 12)$
3. C. $(\frac{24}{5}, 10)$
4. D. $(\frac{20}{3}, 12)$
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Answer: A
Q252020 · 3 Sept 2020 · Shift 2· Numerical answer
If the tangent to the curve, $y = e^{x}$ at a point $(c, e^{c})$ and the normal to the parabola, $y^{2} = 4 x$ at the point $(1, 2)$ intersect at the same point on the x-axis, then the value of c is _______.
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Answer: 4
Q42020 · 2 Sept 2020 · Shift 1
Area (in sq. units) of the region outside $\frac{∣ x ∣}{2} + \frac{∣ y ∣}{3} = 1$ and inside the ellipse $\frac{x ^{2}}{4} + \frac{y ^{2}}{9} = 1$ is :
1. A. $6 (4 - π)$
2. B. $3 (4 - π)$
3. C. $6 (π - 2)$
4. D. $3 (π - 2)$
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Answer: C
Q202020 · 2 Sept 2020 · Shift 1
A line parallel to the straight line $2 x - y = 0$ is tangent to the hyperbola $\frac{x ^{2}}{4} - \frac{y ^{2}}{2} = 1$ at the point $(x_{1}, y_{1})$ . Then $x_{1}^{2} + 5 y_{1}^{2}$ is equal to :
1. A.5
2. B.6
3. C.10
4. D.8
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Answer: B
Q12020 · 2 Sept 2020 · Shift 2
The area (in sq. units) of an equilateral triangle inscribed in the parabola $y^{2} = 8 x$ , with one of its vertices on the vertex of this parabola, is :
1. A. $2563$
2. B. $643$
3. C. $1283$
4. D. $1923$
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Answer: D
Q102020 · 2 Sept 2020 · Shift 2
For some $θ \in (0, \frac{π}{2})$ , if the eccentricity of the hyperbola, $x^{2} - y^{2} sec^{2} θ = 10$ is $5$ times the eccentricity of the ellipse, $x^{2} sec^{2} θ + y^{2} = 5$ , then the length of the latus rectum of the ellipse, is :
1. A. $30$
2. B. $26$
3. C. $\frac{4 5}{3}$
4. D. $\frac{2 5}{3}$
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Answer: C
Q92020 · 9 Jan 2020 · Shift 1
If $e_{1}$ and $e_{2}$ are the eccentricities of the ellipse, $\frac{x ^{2}}{18} + \frac{y ^{2}}{4} = 1$ and the hyperbola, $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ respectively and $(e_{1}, e_{2})$ is a point on the ellipse, $15 x^{2} + 3 y^{2} = k$ , then $k$ is equal to :
1. A.17
2. B.16
3. C.15
4. D.14
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Answer: B
Q172020 · 9 Jan 2020 · Shift 2
The length of the minor axis (along y-axis) of an ellipse in the standard form is $\frac{4}{3}$ . If this ellipse touches the line, $x + 6 y = 8$ ; then its eccentricity is :
1. A. $\frac{1}{3} \frac{11}{3}$
2. B. $\frac{1}{2} \frac{5}{3}$
3. C. $\frac{5}{6}$
4. D. $\frac{1}{2} \frac{11}{3}$
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Answer: D
Q182020 · 9 Jan 2020 · Shift 2
If one end of a focal chord AB of the parabola $y^{2} = 8 x$ is at $A (\frac{1}{2}, - 2)$ , then the equation of the tangent to it at B is :
1. A. $2 x - y - 24 = 0$
2. B. $x - 2 y + 8 = 0$
3. C. $x + 2 y + 8 = 0$
4. D. $2 x + y - 24 = 0$
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Answer: B
Q102020 · 8 Jan 2020 · Shift 1
The locus of a point which divides the line segment joining the point $(0, - 1)$ and a point on the parabola, $x^{2} = 4 y$ , internally in the ratio $1 : 2$ , is :
1. A. $9 x^{2} - 3 y = 2$
2. B. $4 x^{2} - 3 y = 2$
3. C. $x^{2} - 3 y = 2$
4. D. $9 x^{2} - 12 y = 8$
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Answer: D
Q172020 · 8 Jan 2020 · Shift 1
Let the line $y = m x$ and the ellipse $2 x^{2} + y^{2} = 1$ intersect at a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at $(- \frac{1}{3 2}, 0)$ and $(0, β)$ , then $β$ is equal to :
1. A. $\frac{2}{3}$
2. B. $\frac{2}{3}$
3. C. $\frac{2 2}{3}$
4. D. $\frac{2}{3}$
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Answer: A
Q182020 · 8 Jan 2020 · Shift 2
If a hyperbola passes through the point P(10, 16) and it has vertices at (± 6, 0), then the equation of the normal to it at P is :
1. A.2x + 5y = 100
2. B.x + 3y = 58
3. C.x + 2y = 42
4. D.3x + 4y = 94
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Answer: A
Q242020 · 8 Jan 2020 · Shift 2· Numerical answer
Let a line $y = m x$ ( $m > 0$ ) intersect the parabola, $y^{2} = x$ at a point P, other than the origin. Let the tangent to it at P meet the x-axis at the point Q. If area $(Δ OPQ) = 4 sq. units$ , then m is equal to _______.
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Answer: 0.5
Q72020 · 7 Jan 2020 · Shift 1
If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of its latus rectum is :
1. A. $3$
2. B. $32$
3. C. $\frac{3}{2}$
4. D. $23$
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Answer: B
Q82020 · 7 Jan 2020 · Shift 1
The area of the region, enclosed by the circle $x^{2} + y^{2} = 2$ which is not common to the region bounded by the parabola $y^{2} = x$ and the straight line $y = x$ , is:
1. A. $\frac{1}{6} (24 π - 1)$
2. B. $\frac{1}{3} (12 π - 1)$
3. C. $\frac{1}{3} (6 π - 1)$
4. D. $\frac{1}{6} (12 π - 1)$
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Answer: D
Q172020 · 7 Jan 2020 · Shift 1
If $y = m x + 4$ is a tangent to both the parabolas, $y^{2} = 4 x$ and $x^{2} = 2 b y$ , then b is equal to :
1. A.-128
2. B.128
3. C.-64
4. D.-32
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Answer: A
Q152020 · 7 Jan 2020 · Shift 2
If $3 x + 4 y = 122$ is a tangent to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{9} = 1$ for some $a \in R$ , then the distance between the foci of the ellipse is :
1. A. $25$
2. B. $27$
3. C.4
4. D. $22$
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Answer: B