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Friday, 9 October

JEE Main Maths · Class 11

Conic Sections: JEE Main previous year questions

Conic Sections had 116 questions in the 42 JEE Main shifts Lumi analysed, about 2.4 a shift, asked in 41 of 42 shifts. It ranks 1 of 22 Maths chapters by questions; 38% of its questions were hard.

Questions
116
in 41 of 42 shifts
A shift, on average
2.4
of 25 Maths questions
Since 2024
+0.6
2.2 a shift in 2017–23, 2.8 in 2024–26
Numerical answer
34
29% of its questions

Year by year

How often it came

Questions in the analysed shifts of each year. Years differ in how many shifts Lumi analysed, so read the bars with the shift count.

Questions each year
014282’179’2012’2112’229’2318’2426’2528’26

2017: 1 shifts · 2020: 4 shifts · 2021: 4 shifts · 2022: 4 shifts · 2023: 4 shifts · 2024: 8 shifts · 2025: 8 shifts · 2026: 9 shifts

Difficulty by year
  • 20172
  • 20209
  • 202112
  • 202212
  • 20239
  • 202418
  • 202526
  • 202628
  • Easy
  • Medium
  • Hard

Inside the chapter

Topics and the ideas that repeat

13 topics and 43 distinct ideas; 16 ideas were asked more than once. Most questions are single correct and multi-step.

Topics, by questions

45 questions had no close topic in our taxonomy and are counted for the chapter only.

Ideas asked most
16 repeat

Study it on Lumi

Conic Sections: the interactive lesson

Every question

All 116 Conic Sections questions

Newest first. Each opens with its options and the official answer.

2026 28 questions

  1. 2 Apr 2026, Shift 1 · Q10Let an ellipse x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a<ba<b, pass through the point (4,3)(4, 3) and have eccentricity 53\frac{\sqrt5}{3}. Then…MediumSingle correct
  2. 2 Apr 2026, Shift 1 · Q24Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts…HardNumerical value
  3. 2 Apr 2026, Shift 2 · Q10Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines…MediumSingle correct
  4. 2 Apr 2026, Shift 2 · Q11Let OO be the origin, and PP and QQ be two points on the rectangular hyperbola xy=12xy = 12 such that the midpoint of the line segment…MediumSingle correct
  5. 2 Apr 2026, Shift 2 · Q12Let the parabola y=x2+px+qy = x^2 + px + q passing through the point (1,−1)(1, -1) be such that the distance between its vertex and the xx-axis is…MediumSingle correct
  6. 2 Apr 2026, Shift 2 · Q23Let AA be the point (3,0)(3, 0) and circles with variable diameter ABAB touch the circle x2+y2=36x^2 + y^2 = 36 internally. Let the curve CC be…HardNumerical value
  7. 4 Apr 2026, Shift 1 · Q13Suppose that two chords, drawn from the point (1,2)(1,2) on the circle x2+y2+x−3y=0x^2+y^2+x-3y=0 are bisected by the yy-axis. If the other ends of…HardSingle correct
  8. 4 Apr 2026, Shift 1 · Q22Consider the parabola P:y2=4kxP: y^2=4kx and the ellipse E:x2a2+y2b2=1E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. Let the line segment joining the points of…MediumNumerical value
  9. 4 Apr 2026, Shift 2 · Q10Let P(3cos⁡α,2sin⁡α)P(3\cos\alpha,2\sin\alpha), α≠0\alpha\neq0, be a point on the ellipse x29+y24=1\frac{x^2}{9}+\frac{y^2}{4}=1, QQ be a point on the circle…HardSingle correct
  10. 4 Apr 2026, Shift 2 · Q11Let H:x2a2−y2b2=1H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 be a hyperbola such that the distance between its foci is 6 and the distance between its…MediumSingle correct
  11. 4 Apr 2026, Shift 2 · Q24Let A,BA,B and CC be the vertices of a variable right angled triangle inscribed in the parabola y2=16xy^2=16x. Let the vertex BB containing…HardNumerical value
  12. 5 Apr 2026, Shift 1 · Q9Let a focus of the ellipse E:x2a2+y2b2=1E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 be S(4,0)S(4, 0) and its eccentricity be 45\frac{4}{5}. If the point…EasySingle correct
  13. 5 Apr 2026, Shift 1 · Q10Let P be a moving point on the circle x2+y2−6x−8y+21=0x^2 + y^2 - 6x - 8y + 21 = 0. Then, the maximum distance of P from the vertex of the parabola…MediumSingle correct
  14. 5 Apr 2026, Shift 2 · Q11Let the point P be the vertex of the parabola y=x2−6x+12y=x^2-6x+12. If a line passing through the point P intersects the circle…HardSingle correct
  15. 5 Apr 2026, Shift 2 · Q12Let the directrix of the parabola P:y2=8xP: y^2=8x, cut x-axis at the point A. Let B(α,β)B(\alpha, \beta), α>1\alpha>1, be a point on P such that…MediumSingle correct
  16. 5 Apr 2026, Shift 2 · Q13Let the eccentricity e of a hyperbola satisfy the equation 6e2−11e+3=06e^2-11e+3=0. If the foci of the hyperbola are (3,5)(3, 5) and (3,−4)(3, -4), then…MediumSingle correct
  17. 6 Apr 2026, Shift 1 · Q2Let one root of the quadratic equation in xx: (k2−15k+27)x2+9(k−1)x+18=0(k^2-15k+27)x^2+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of…MediumSingle correct
  18. 6 Apr 2026, Shift 1 · Q3Let e1e_1 and e2e_2 be two distinct roots of the equation x2−ax+2=0x^2-ax+2=0. Let the sets {a∈R:e1\{a\in\mathbb{R}: e_1 and e2e_2 are the…HardSingle correct
  19. 6 Apr 2026, Shift 1 · Q11If the eccentricity ee of the hyperbola x2a2−y2b2=1\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, passing through (6,43)\left(6,4\sqrt{3}\right), satisfies…MediumSingle correct
  20. 6 Apr 2026, Shift 1 · Q12Let chord PQ of length 3133\sqrt{13} of the parabola y2=12xy^2=12x be such that the ordinates of points P and Q are in the ratio 1:2. If the…HardSingle correct
  21. 6 Apr 2026, Shift 1 · Q22Let the centre of the circle x2+y2+2gx+2fy+25=0x^2+y^2+2gx+2fy+25=0 be in the first quadrant and lie on the line 2x−y=42x-y=4. Let the area of an equilateral…MediumNumerical value
  22. 6 Apr 2026, Shift 2 · Q10Let C be a circle having centre in the first quadrant and touching the xx-axis at a distance of 3 units from the origin. If the circle C…MediumSingle correct
  23. 6 Apr 2026, Shift 2 · Q11The eccentricity of an ellipse E with centre at the origin O is 32\frac{\sqrt{3}}{2} and its directrices are x=±463x=\pm\frac{4\sqrt{6}}{3}.…MediumSingle correct
  24. 6 Apr 2026, Shift 2 · Q12Let x=9x=9 be a directrix of an ellipse E, whose centre is at the origin and eccentricity is 13\frac{1}{3}. Let P (α,0)(\alpha,0), α>0\alpha>0,…MediumSingle correct
  25. 6 Apr 2026, Shift 2 · Q23Let the line x−y=4x-y=4 intersect the circle C : (x−4)2+(y+3)2=9(x-4)^2+(y+3)^2=9 at the points Q and R. If P(α,β)(\alpha,\beta) is a point on C such that PQ…MediumNumerical value
  26. 8 Apr 2026, Shift 2 · Q11Let O be the vertex of the parabola y2=4xy^2 = 4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of…MediumSingle correct
  27. 8 Apr 2026, Shift 2 · Q20Let x2f(a2+7a+3)+y2f(3a+15)=1\frac{x^2}{f(a^2 + 7a + 3)} + \frac{y^2}{f(3a + 15)} = 1 represent an ellipse with major axis along yy-axis, where ff is a…HardSingle correct
  28. 8 Apr 2026, Shift 2 · Q24Consider the circle C:x2+y2−6x−8y−11=0C : x^2 + y^2 - 6x - 8y - 11 = 0. Let a variable chord AB of the circle C subtend a right angle at the origin. If…HardNumerical value

2025 26 questions

  1. 2 Apr 2025, Shift 1 · Q1Let the focal chord PQ of the parabola y2=4xy^2=4x make an angle of 60∘60^\circ with the positive xx-axis, where P lies in the first…MediumSingle correct
  2. 2 Apr 2025, Shift 1 · Q4If S and S' are the foci of the ellipse x218+y29=1\frac{x^2}{18}+\frac{y^2}{9}=1 and P be a point on the ellipse, then…MediumSingle correct
  3. 2 Apr 2025, Shift 1 · Q10Let one focus of the hyperbola H:x2a2−y2b2=1\mathrm{H}: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 be at (10,0)(\sqrt{10},0) and the corresponding directrix be…EasySingle correct
  4. 2 Apr 2025, Shift 1 · Q22The absolute difference between the squares of the radii of the two circles passing through the point (−9,4)(-9, 4) and touching the lines…HardNumerical value
  5. 2 Apr 2025, Shift 2 · Q11Let the point PP of the focal chord PQPQ of the parabola y2=16xy^2 = 16x be (1,−4)(1, -4). If the focus of the parabola divides the chord PQPQ in…MediumSingle correct
  6. 2 Apr 2025, Shift 2 · Q12If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the…EasySingle correct
  7. 3 Apr 2025, Shift 1 · Q11The radius of the smallest circle which touches the parabolas y=x2+2y = x^2 + 2 and x=y2+2x = y^2 + 2 isHardSingle correct
  8. 3 Apr 2025, Shift 1 · Q12A line passing through the point P(5,5)P(\sqrt{5}, \sqrt{5}) intersects the ellipse x236+y225=1\frac{x^2}{36} + \frac{y^2}{25} = 1 at AA and BB such…HardSingle correct
  9. 3 Apr 2025, Shift 1 · Q23Let the product of the focal distances of the point P(4,23)\mathrm{P}\left(4, 2\sqrt{3}\right) on the hyperbola…MediumNumerical value
  10. 3 Apr 2025, Shift 2 · Q11If the four distinct points (4,6),(−1,5),(0,0)(4, 6), (-1, 5), (0, 0) and (k,3k)(k, 3k) lie on a circle of radius rr, then 10k+r210k + r^2 is equal toMediumSingle correct
  11. 3 Apr 2025, Shift 2 · Q12Let CC be the circle of minimum area enclosing the ellipse E:x2a2+y2b2=1E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 with eccentricity 12\frac{1}{2} and…HardSingle correct
  12. 3 Apr 2025, Shift 2 · Q23If the equation of the hyperbola with foci (4,2)(4, 2) and (8,2)(8, 2) is 3x2−y2−αx+βy+γ=03x^2 - y^2 - \alpha x + \beta y + \gamma = 0, then…MediumNumerical value
  13. 4 Apr 2025, Shift 2 · Q9The axis of a parabola is the line y=xy=x and its vertex and focus are in the first quadrant at distances 2\sqrt{2} and 222\sqrt{2} units…HardSingle correct
  14. 4 Apr 2025, Shift 2 · Q10Let for two distinct values of p the lines y=x+py=x+p touch the ellipse E:x242+y232=1E:\frac{x^2}{4^2}+\frac{y^2}{3^2}=1 at the points A and B. Let the…HardSingle correct
  15. 4 Apr 2025, Shift 2 · Q11The centre of a circle C is at the centre of the ellipse E:x2a2+y2b2=1E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>ba>b. Let C pass through the foci F1F_1…HardSingle correct
  16. 4 Apr 2025, Shift 2 · Q12Let the sum of the focal distances of the point P(4,3)P(4,3) on the hyperbola H:x2a2−y2b2=1H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 be 8538\sqrt{\frac{5}{3}}.…HardSingle correct
  17. 4 Apr 2025, Shift 2 · Q19A line passing through the point A(−2,0)A(-2,0), touches the parabola P:y2=x−2P:y^2=x-2 at the point B in the first quadrant. The area, of the region…MediumSingle correct
  18. 7 Apr 2025, Shift 1 · Q10Let P be the parabola, whose focus is (−2,1)(-2, 1) and directrix is 2x+y+2=02x + y + 2 = 0. Then the sum of the ordinates of the points on P, whose…MediumSingle correct
  19. 7 Apr 2025, Shift 1 · Q12Let C1C_1 be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let C2C_2 be the circle with centre (1,3)(1, 3)…MediumSingle correct
  20. 7 Apr 2025, Shift 1 · Q23Consider the hyperbola x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 having one of its focus at P(−3,0)P(-3, 0). If the latus ractum through its…HardNumerical value
  21. 7 Apr 2025, Shift 2 · Q1Let A={(α,β)∈R×R:∣α−1∣≤4 and ∣β−5∣≤6}A=\{(\alpha,\beta)\in\mathbf{R}\times\mathbf{R}: |\alpha-1|\le 4 \text{ and } |\beta-5|\le 6\} and…MediumSingle correct
  22. 7 Apr 2025, Shift 2 · Q12Let the length of a latus rectum of an ellipse x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 be 10. If its eccentricity is the minimum value of the…MediumSingle correct
  23. 7 Apr 2025, Shift 2 · Q13Let e1e_1 and e2e_2 be the eccentricities of the ellipse x2b2+y225=1\frac{x^2}{b^2}+\frac{y^2}{25}=1 and the hyperbola…MediumSingle correct
  24. 7 Apr 2025, Shift 2 · Q22Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2a2a and 2b2b, respectively, and one focus and the…HardNumerical value
  25. 8 Apr 2025, Shift 2 · Q12Let the ellipse 3x2+py2=43x^2 + py^2 = 4 pass through the centre CC of the circle x2+y2−2x−4y−11=0x^2 + y^2 - 2x - 4y - 11 = 0 of radius rr. Let f1,f2f_1, f_2…MediumSingle correct
  26. 8 Apr 2025, Shift 2 · Q23Let rr be the radius of the circle, which touches xx-axis at point (a,0)(a, 0), a<0a < 0 and the parabola y2=9xy^2 = 9x at the point (4,6)(4, 6).…HardNumerical value

2024 18 questions

  1. 8 Apr 2024, Shift 1 · Q15Let the circles C1:(x−α)2+(y−β)2=r12C_1:(x-\alpha)^2+(y-\beta)^2=r_1^2 and C2:(x−8)2+(y−152)2=r22C_2:(x-8)^2+\left(y-\frac{15}{2}\right)^2=r_2^2 touch each other externally at…MediumSingle correct
  2. 8 Apr 2024, Shift 1 · Q16Let H:−x2a2+y2b2=1H:\frac{-x^2}{a^2}+\frac{y^2}{b^2}=1 be the hyperbola, whose eccentricity is 3\sqrt{3} and the length of the latus rectum is…MediumSingle correct
  3. 9 Apr 2024, Shift 1 · Q12Let f(x)=x2+9f(x)=x^2+9, g(x)=xx−9g(x)=\frac{x}{x-9} and a=f∘g(10)a=f\circ g(10), b=g∘f(3)b=g\circ f(3). If ee and ll denote the eccentricity and the length of the…EasySingle correct
  4. 9 Apr 2024, Shift 1 · Q13Let a circle passing through (2,0)(2,0) have its centre at the point (h,k)(h,k). Let (xc,yc)(x_c,y_c) be the point of intersection of the lines…MediumSingle correct
  5. 9 Apr 2024, Shift 1 · Q29Let the centre of a circle, passing through the points (0,0)(0,0), (1,0)(1,0) and touching the circle x2+y2=9x^2+y^2=9, be (h,k)(h,k). Then for all…HardNumerical value
  6. 9 Apr 2024, Shift 2 · Q15Let the foci of a hyperbola HH coincide with the foci of the ellipse E:(x−1)2100+(y−1)275=1E:\frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1 and the eccentricity of…MediumSingle correct
  7. 9 Apr 2024, Shift 2 · Q27Consider the circle C:x2+y2=4C: x^2+y^2=4 and the parabola P:y2=8xP: y^2=8x. If the set of all values of α\alpha, for which three chords of the…HardNumerical value
  8. 9 Apr 2024, Shift 2 · Q28Let AA, BB and CC be three points on the parabola y2=6xy^2=6x and let the line segment ABAB meet the line LL through CC parallel to the…HardNumerical value
  9. 29 Jan 2024, Shift 1 · Q27Equations of two diameters of a circle are 2x−3y=52x-3y=5 and 3x−4y=73x-4y=7. The line joining the points (−227,−4)\left(-\frac{22}{7},-4\right) and…MediumNumerical value
  10. 29 Jan 2024, Shift 1 · Q28If the points of intersection of two distinct conics x2+y2=4bx^2+y^2=4b and x216+y2b2=1\frac{x^2}{16}+\frac{y^2}{b^2}=1 lie on the curve y2=3x2y^2=3x^2, then…HardNumerical value
  11. 30 Jan 2024, Shift 2 · Q14Let A(α,0)A(\alpha, 0) and B(0,β)B(0, \beta) be the points on the line 5x+7y=505x + 7y = 50. Let the point PP divide the line segment ABAB internally…MediumSingle correct
  12. 30 Jan 2024, Shift 2 · Q15Let PP be a point on the hyperbola H:x29−y24=1H: \frac{x^2}{9} - \frac{y^2}{4} = 1, in the first quadrant such that the area of triangle formed by…MediumSingle correct
  13. 30 Jan 2024, Shift 2 · Q28Consider two circles C1:x2+y2=25C_1: x^2 + y^2 = 25 and C2:(x−α)2+y2=16C_2: (x - \alpha)^2 + y^2 = 16, where α∈(5,9)\alpha \in (5, 9). Let the angle between the two…HardNumerical value
  14. 31 Jan 2024, Shift 1 · Q13If one of the diameters of the circle x2+y2−10x+4y+13=0x^2+y^2-10x+4y+13=0 is a chord of another circle C, whose center is the point of intersection of…MediumSingle correct
  15. 31 Jan 2024, Shift 1 · Q16If the foci of a hyperbola are same as that of the ellipse x29+y225=1\frac{x^2}{9}+\frac{y^2}{25}=1 and the eccentricity of the hyperbola is…HardSingle correct
  16. 31 Jan 2024, Shift 1 · Q28Let the foci and length of the latus rectum of an ellipse x2a2+y2b2=1, a>b\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\ a>b be (±5,0)(\pm5,0) and 50\sqrt{50},…MediumNumerical value
  17. 31 Jan 2024, Shift 2 · Q10Let a variable line passing through the centre of the circle x2+y2−16x−4y=0x^2+y^2-16x-4y=0, meet the positive co-ordinate axes at the points AA and…MediumSingle correct
  18. 31 Jan 2024, Shift 2 · Q14Let PP be a parabola with vertex (2,3)(2,3) and directrix 2x+y=62x+y=6. Let an ellipse E:x2a2+y2b2=1, a>bE:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\ a>b, of…HardSingle correct

2023 9 questions

  1. 6 Apr 2023, Shift 1 · Q27A circle passing through the point P(α,β)P(\alpha,\beta) in the first quadrant touches the two coordinate axes at the points A and B. The…HardNumerical value
  2. 6 Apr 2023, Shift 1 · Q28Let the tangent to the curve x2+2x−4y+9=0x^2+2x-4y+9=0 at the point P(1,3)P(1,3) on it meet the yy-axis at A. Let the line passing through P and…MediumNumerical value
  3. 8 Apr 2023, Shift 2 · Q11Let OO be the origin and OPOP and OQOQ be the tangents to the circle x2+y2−6x+4y+8=0x^2 + y^2 - 6x + 4y + 8 = 0 at the points PP and QQ on it. If…MediumSingle correct
  4. 8 Apr 2023, Shift 2 · Q12Let A(0,1)A(0, 1), B(1,1)B(1, 1) and C(1,0)C(1, 0) be the mid-points of the sides of a triangle with incentre at the point DD. If the focus of the…HardSingle correct
  5. 8 Apr 2023, Shift 2 · Q29The ordinates of the points PP and QQ on the parabola with focus (3,0)(3, 0) and directrix x=−3x = -3 are in the ratio 3:13:1. If…HardNumerical value
  6. 11 Apr 2023, Shift 1 · Q3Consider ellipses Ek:kx2+k2y2=1, k=1,2,…,20E_k : kx^2 + k^2y^2 = 1,\ k = 1, 2, \ldots, 20. Let CkC_k be the circle which touches the four chords joining the end…HardSingle correct
  7. 11 Apr 2023, Shift 1 · Q28Let Hn:x21+n−y23+n=1, n∈NH_n : \frac{x^2}{1+n} - \frac{y^2}{3+n} = 1,\ n \in \mathbb{N}. Let kk be the smallest even value of nn such that the eccentricity…MediumNumerical value
  8. 13 Apr 2023, Shift 2 · Q13Let the centre of a circle C be (α,β)(\alpha,\beta) and its radius r<8r<8. Let 3x+4y=243x+4y=24 and 3x−4y=323x-4y=32 be two tangents and 4x+3y=14x+3y=1 be a…MediumSingle correct
  9. 13 Apr 2023, Shift 2 · Q28The foci of a hyperbola are (±2,0)(\pm2,0) and its eccentricity is 32\frac{3}{2}. A tangent, perpendicular to the line 2x+3y=62x+3y=6, is drawn at…HardNumerical value

2022 12 questions

  1. 25 Jul 2022, Shift 1 · Q13Let the locus of the centre (α,β)(\alpha,\beta), β>0\beta>0, of the circle which touches the circle x2+(y−1)2=1x^2+(y-1)^2=1 externally and also…MediumSingle correct
  2. 25 Jul 2022, Shift 1 · Q28Let the equation of two diameters of a circle x2+y2−2x+2fy+1=0x^2+y^2-2x+2fy+1=0 be 2px−y=12px-y=1 and 2x+py=4p2x+py=4p. Then the slope m∈(0,∞)m\in(0,\infty) of the…MediumNumerical value
  3. 25 Jul 2022, Shift 1 · Q29The sum of diameters of the circles that touch (i) the parabola 75x2=64(5y−3)75x^2=64(5y-3) at the point (85,65)\left(\frac{8}{5},\frac{6}{5}\right) and…HardNumerical value
  4. 28 Jul 2022, Shift 1 · Q11If the tangents drawn at the points P and Q on the parabola y2=2x−3y^2=2x-3 intersect at the point R(0, 1), then the orthocentre of the…HardSingle correct
  5. 28 Jul 2022, Shift 1 · Q12Let C be the centre of the circle x2+y2−x+2y=114x^2+y^2-x+2y=\frac{11}{4} and P be a point on the circle. A line passes through the point C, makes an…MediumSingle correct
  6. 28 Jul 2022, Shift 1 · Q27For the hyperbola H:x2−y2=1H:x^2-y^2=1 and the ellipse E:x2a2+y2b2=1, a>b>0E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\ a>b>0, let the (1) eccentricity of E be reciprocal…HardNumerical value
  7. 24 Jun 2022, Shift 1 · Q5Let x2+y2+Ax+By+C=0x^2+y^2+Ax+By+C=0 be a circle passing through (0,6)(0, 6) and touching the parabola y=x2y=x^2 at (2,4)(2, 4). Then A+CA+C is equal to ______.HardSingle correct
  8. 24 Jun 2022, Shift 1 · Q15Let λx−2y=μ\lambda x-2y=\mu be a tangent to the hyperbola a2x2−y2=b2a^2x^2-y^2=b^2. Then…MediumSingle correct
  9. 24 Jun 2022, Shift 1 · Q28If two tangents drawn from a point (α,β)(\alpha, \beta) lying on the ellipse 25x2+4y2=125x^2+4y^2=1 to the parabola y2=4xy^2=4x are such that the slope…HardNumerical value
  10. 29 Jun 2022, Shift 1 · Q18Let PQ be a focal chord of the parabola y2=4xy^2 = 4x such that it subtends an angle of π2\frac{\pi}{2} at the point (3,0)(3, 0). Let the line…HardSingle correct
  11. 29 Jun 2022, Shift 1 · Q19Let the tangent to the circle C1:x2+y2=2C_1: x^2 + y^2 = 2 at the point M(−1,1)M(-1, 1) intersect the circle C2:(x−3)2+(y−2)2=5C_2: (x - 3)^2 + (y - 2)^2 = 5, at two…HardSingle correct
  12. 29 Jun 2022, Shift 1 · Q28Let H:x2a2−y2b2=1H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, a>0,b>0a > 0, b > 0, be a hyperbola such that the sum of lengths of the transverse and the…MediumNumerical value

2021 12 questions

  1. 3 Aug 2021, Shift 2 · Q74If an ellipse passes through the point (4,1)(4, 1) and has foci at (±3,0)(\pm3, 0), then its eccentricity isMediumSingle correct
  2. 3 Aug 2021, Shift 2 · Q75The tangents, to the parabola y2=2−xy^2 = 2 - x at the points of its intersection with the line y=x−2y = x - 2, intersect at the pointMediumSingle correct
  3. 3 Aug 2021, Shift 2 · Q86The circles, x2+y2+2x+4y−4=0x^2 + y^2 + 2x + 4y - 4 = 0 and x2+y2−4x+4y+k=0x^2 + y^2 - 4x + 4y + k = 0 touch each other internally. If the point of contact of these…MediumNumerical value
  4. 25 Jul 2021, Shift 1 · Q61Let an ellipse E:x2a2+y2b2=1E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a2>b2a^2>b^2, passes through (32,1)\left(\sqrt{\frac{3}{2}},1\right) and has eccentricity…HardSingle correct
  5. 25 Jul 2021, Shift 1 · Q68Let a parabola P be such that its vertex and focus lie on the positive xx-axis at a distance 2 and 4 units from the origin, respectively.…MediumSingle correct
  6. 25 Jul 2021, Shift 1 · Q75The locus of the centroid of the triangle formed by any point P on the hyperbola 16x2−9y2+32x+36y−164=016x^2-9y^2+32x+36y-164=0, and its foci is :HardSingle correct
  7. 16 Mar 2021, Shift 2 · Q62Let the lengths of intercepts on xx-axis and yy-axis made by the circle x2+y2+ax+2ay+c=0x^2+y^2+ax+2ay+c=0, (a<0a<0) be 222\sqrt2 and 252\sqrt5,…MediumSingle correct
  8. 16 Mar 2021, Shift 2 · Q65If the points of intersections of the ellipse x216+y2b2=1\frac{x^2}{16}+\frac{y^2}{b^2}=1 and the circle x2+y2=4bx^2+y^2=4b, b>4b>4 lie on the curve…MediumSingle correct
  9. 16 Mar 2021, Shift 2 · Q66Let C be the locus of the mirror image of a point on the parabola y2=4xy^2=4x with respect to the line y=xy=x. Then the equation of tangent to…MediumSingle correct
  10. 18 Mar 2021, Shift 1 · Q75Choose the correct statement about two circles whose equations are given below : x2+y2−10x−10y+41=0x^2+y^2-10x-10y+41=0 x2+y2−22x−10y+137=0x^2+y^2-22x-10y+137=0EasySingle correct
  11. 18 Mar 2021, Shift 1 · Q76For the four circles M, N, O and P, following four equations are given : Circle M : x2+y2=1x^2+y^2=1 Circle N : x2+y2−2x=0x^2+y^2-2x=0 Circle O :…EasySingle correct
  12. 18 Mar 2021, Shift 1 · Q84A square ABCD has all its vertices on the curve x2y2=1x^2y^2=1. The midpoints of its sides also lie on the same curve. Then, the square of…HardNumerical value

2020 9 questions

  1. 8 Jan 2020, Shift 1 · Q63The locus of a point which divides the line segment joining the point (0,−1)(0,-1) and a point on the parabola, x2=4yx^2=4y, internally in the…MediumSingle correct
  2. 8 Jan 2020, Shift 1 · Q64Let the line y=mxy=mx and the ellipse 2x2+y2=12x^2+y^2=1 intersect at a point P in the first quadrant. If the normal to this ellipse at P meets…MediumSingle correct
  3. 9 Jan 2020, Shift 2 · Q65The length of the minor axis (along yy-axis) of an ellipse in the standard form is 43\frac{4}{\sqrt3}. If this ellipse touches the line,…MediumSingle correct
  4. 9 Jan 2020, Shift 2 · Q66If one end of a focal chord AB of the parabola y2=8xy^2=8x is at A(12,−2)A\left(\frac{1}{2},-2\right), then the equation of the tangent to it at B…MediumSingle correct
  5. 9 Jan 2020, Shift 2 · Q73If the curves, x2−6x+y2+8=0x^2-6x+y^2+8=0 and x2−8y+y2+16−k=0x^2-8y+y^2+16-k=0, (k>0)(k>0) touch each other at a point, then the largest value of kk is ______.MediumNumerical value
  6. 2 Sep 2020, Shift 1 · Q65A line parallel to the straight line 2x−y=02x-y=0 is tangent to the hyperbola x24−y22=1\frac{x^{2}}{4}-\frac{y^{2}}{2}=1 at the point (x1,y1)(x_1, y_1).…MediumSingle correct
  7. 2 Sep 2020, Shift 1 · Q74The number of integral values of k for which the line, 3x+4y=k3x+4y=k intersects the circle, x2+y2−2x−4y+4=0x^{2}+y^{2}-2x-4y+4=0 at two distinct points is…MediumNumerical value
  8. 6 Sep 2020, Shift 2 · Q65The centre of the circle passing through the point (0,1)(0,1) and touching the parabola y=x2y=x^2 at the point (2,4)(2,4) is :HardSingle correct
  9. 6 Sep 2020, Shift 2 · Q66If 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…MediumSingle correct

2017 2 questions

  1. 2 Apr 2017 (offline) · Q80The eccentricity of an ellipse whose centre is at the origin is 12\frac{1}{2}. If one of its directrices is x=−4x=-4, then the equation of…MediumSingle correct
  2. 2 Apr 2017 (offline) · Q81A hyperbola passes through the point P(2,3)P(\sqrt{2},\sqrt{3}) and has foci at (±2,0)(\pm2,0). Then the tangent to this hyperbola at P also…MediumSingle correct

Pattern: Asked most years.