Conic Sections · JEE Main previous year questions

242 questions asked across 19 exam years, newest first

Year-wise papers Topic-wise questions

Practise this topic

2021 15 questions

Q182021 · 1 Sept 2021 · Shift 2
Consider the parabola with vertex $(\frac{1}{2}, \frac{3}{4})$ and the directrix $y = \frac{1}{2}$ . Let P be the point where the parabola meets the line $x = - \frac{1}{2}$ . If the normal to the parabola at P intersects the parabola again at the point Q, then $(PQ)^{2}$ is equal to :
1. A. $\frac{75}{8}$
2. B. $\frac{125}{16}$
3. C. $\frac{25}{2}$
4. D. $\frac{15}{2}$
Show answer
Answer: B
Q102021 · 26 Aug 2021 · Shift 2
The locus of the mid points of the chords of the hyperbola $x^{2} - y^{2} = 4$ , which touch the parabola $y^{2} = 8 x$ , is :
1. A. $y^{3} (x - 2) = x^{2}$
2. B. $x^{3} (x - 2) = y^{2}$
3. C. $y^{2} (x - 2) = x^{3}$
4. D. $x^{2} (x - 2) = y^{3}$
Show answer
Answer: C
Q172021 · 26 Aug 2021 · Shift 2
The point $P (- 26, 3)$ lies on the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ having eccentricity $\frac{5}{2}$ . If the tangent and normal at P to the hyperbola intersect its conjugate axis at the point Q and R respectively, then QR is equal to :
1. A. $43$
2. B. $6$
3. C. $63$
4. D. $36$
Show answer
Answer: C
Q302021 · 27 Jul 2021 · Shift 2· Numerical answer
Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, $- 4$ ), one focus at (4, $- 4$ ) and one vertex at (5, $- 4$ ). If $m x - y = 4$ , m > 0 is a tangent to the ellipse E, then the value of $5 m^{2}$ is equal to ____.
Show answer
Answer: 3
Q32021 · 25 Jul 2021 · Shift 1
The locus of the centroid of the triangle formed by any point P on the hyperbola $16 x^{2} - 9 y^{2} + 32 x + 36 y - 164 = 0$ , and its foci is :
1. A. $16 x^{2} - 9 y^{2} + 32 x + 36 y - 36 = 0$
2. B. $9 x^{2} - 16 y^{2} + 36 x + 32 y - 144 = 0$
3. C. $16 x^{2} - 9 y^{2} + 32 x + 36 y - 144 = 0$
4. D. $9 x^{2} - 16 y^{2} + 36 x + 32 y - 36 = 0$
Show answer
Answer: A
Q192021 · 25 Jul 2021 · Shift 1
Let a parabola P be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. units) of $Δ$ SOR is equal to :
1. A. $162$
2. B.16
3. C.32
4. D. $82$
Show answer
Answer: B
Q192021 · 25 Jul 2021 · Shift 2
If a tangent to the ellipse $x^{2} + 4 y^{2} = 4$ meets the tangents at the extremities of it major axis at B and C, then the circle with BC as diameter passes through the point :
1. A. $(3, 0)$
2. B. $(2, 0)$
3. C.(1, 1)
4. D. $(- 1, 1)$
Show answer
Answer: A
Q142021 · 22 Jul 2021 · Shift 2
Let $E_{1} : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , $a > b$ . Let $E_{2}$ be another ellipse such that it touches the end points of major axis of $E_{1}$ and the foci of $E_{2}$ are the end points of minor axis of $E_{1}$ . If $E_{1}$ and $E_{2}$ have same eccentricities, then its value is :
1. A. $\frac{- 1 + 5}{2}$
2. B. $\frac{- 1 + 8}{2}$
3. C. $\frac{- 1 + 3}{2}$
4. D. $\frac{- 1 + 6}{2}$
Show answer
Answer: A
Q202021 · 22 Jul 2021 · Shift 2
Let a line $L : 2 x + y = k$ , $k > 0$ be a tangent to the hyperbola $x^{2} - y^{2} = 3$ . If L is also a tangent to the parabola $y^{2} = α x$ , then $α$ is equal to :
1. A. $12$
2. B. $- 12$
3. C. $24$
4. D. $- 24$
Show answer
Answer: D
Q202021 · 20 Jul 2021 · Shift 1
Let the tangent to the parabola $S : y^{2} = 2 x$ at the point $P (2, 2)$ meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :
1. A. $\frac{25}{2}$
2. B. $\frac{35}{2}$
3. C. $\frac{15}{2}$
4. D. $25$
Show answer
Answer: A
Q252021 · 20 Jul 2021 · Shift 1· Numerical answer
Let T be the tangent to the ellipse $E : x^{2} + 4 y^{2} = 5$ at the point $P (1, 1)$ . If the area of the region bounded by the tangent T, ellipse E, lines $x = 1$ and $x = 5$ is $α 5 + β + γ cos^{- 1} (\frac{1}{5})$ , then $∣ α + β + γ ∣$ is equal to ________.
Show answer
Answer: 1.25
Q182021 · 18 Mar 2021 · Shift 2
Consider a hyperbola $H : x^{2} - 2 y^{2} = 4$ . Let the tangent at a point $P (4, 6)$ meet the x-axis at Q and latus rectum at $R (x_{1}, y_{1}), x_{1} > 0$ . If F is a focus of H which is nearer to the point P, then the area of $Δ QFR$ is equal to :
1. A. $6 - 1$
2. B. $\frac{7}{6} - 2$
3. C. $46 - 1$
4. D. $46$
Show answer
Answer: B
Q202021 · 18 Mar 2021 · Shift 2
Let a tangent be drawn to the ellipse $\frac{x ^{2}}{27} + y^{2} = 1$ at $(33 cos θ, sin θ)$ where $θ \in (0, \frac{π}{2})$ . Then the value of $θ$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :
1. A. $\frac{π}{6}$
2. B. $\frac{π}{3}$
3. C. $\frac{π}{8}$
4. D. $\frac{π}{4}$
Show answer
Answer: A
Q162021 · 16 Mar 2021 · Shift 1
If the three normals drawn to the parabola, $y^{2} = 2 x$ pass through the point $(a, 0)$ , $a \neq = 0$ , then 'a' must be greater than :
1. A. $\frac{1}{2}$
2. B.1
3. C. $- 1$
4. D. $- \frac{1}{2}$
Show answer
Answer: B
Q162021 · 25 Feb 2021 · Shift 2
A hyperbola passes through the foci of the ellipse $\frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is :
1. A. $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$
2. B. $\frac{x ^{2}}{9} - \frac{y ^{2}}{16} = 1$
3. C. $\frac{x ^{2}}{9} - \frac{y ^{2}}{25} = 1$
4. D. $x^{2} - y^{2} = 9$
Show answer
Answer: B