Mathematics · free practice

JEE Main Coordinate Geometry & Vectors: practice questions with solutions

12 questions on 3D Geometry, Circles, Ellipse, Hyperbola, Parabola, Straight Lines, Vectors. Try each one first, then open the solution.

Tip: solve on paper before opening a solution. Once you open it, that question is counted as practised and won't appear in your HeyGyan tests.
HyperbolaMedium

Q1. The latus rectum of a hyperbola equals half its transverse axis. What is its eccentricity?

  1. √(3/2)
  2. √2
  3. 3/2
  4. √3
Show answer and solution

Answer: (A) √(3/2)

SolutionLatus rectum = 2b²/a and transverse axis = 2a. 2b²/a = a gives b² = a²/2. e = √(1 + b²/a²) = √(3/2).
Common trapEquating the latus rectum to the full transverse axis gives √2.
CirclesEasyNumerical

Q2. What is the length of the tangent from the point (5, 1) to the circle x² + y² − 4x + 2y − 4 = 0?

Numerical value question: type the answer, no options.

Show answer and solution

Answer: 2

SolutionLength = √S₁, where S₁ is the circle expression at the point: 25 + 1 − 20 + 2 − 4 = 4. Length = √4 = 2.
Common trapReporting S₁ = 4 without taking the square root.
3D GeometryMedium

Q3. What is the distance between the parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0?

  1. 3/2
  2. 5/2
  3. 7/2
  4. 13/3
Show answer and solution

Answer: (C) 7/2

SolutionMake the coefficients match: the second plane is 2x + y + 2z + 5/2 = 0. Distance = |−8 − 5/2| / √(4 + 1 + 4) = (21/2)/3 = 7/2.
Common trapUsing the planes without matching coefficients first.
ParabolaMedium

Q4. Which points on the parabola y² = 12x are at a distance of 6 from its focus?

  1. (3, ±6)
  2. (6, ±6√2)
  3. (3, ±3)
  4. (9, ±6√3)
Show answer and solution

Answer: (A) (3, ±6)

SolutionHere a = 3. The focal distance of a point (x, y) is x + a, so x + 3 = 6 gives x = 3 and y² = 36, y = ±6.
Common trapMeasuring from the vertex instead of the focus.
Straight LinesEasy

Q5. What is the area of the triangle formed by the lines y = x, y = −x and y = 4?

  1. 8
  2. 12
  3. 16
  4. 32
Show answer and solution

Answer: (C) 16

SolutionThe vertices are (0, 0), (4, 4) and (−4, 4). Base = 8 (along y = 4), height = 4. Area = ½ × 8 × 4 = 16.
Common trapTaking the base as 4 by using only one side of the y-axis.
VectorsMediumNumerical

Q6. What is the volume of the parallelepiped with edges a = î + ĵ, b = ĵ + k̂ and c = k̂ + î?

Numerical value question: type the answer, no options.

Show answer and solution

Answer: 2

SolutionVolume = |[a b c]| = |det[[1,1,0],[0,1,1],[1,0,1]]| = |1(1 − 0) − 1(0 − 1) + 0| = 2.
Common trapDividing by 6 gives the volume of a tetrahedron, not a parallelepiped.
3D GeometryMedium

Q7. What is the foot of the perpendicular from the origin to the plane 2x + y + 2z = 9?

  1. (2, 1, 2)
  2. (1, 2, 2)
  3. (4, 2, 4)
  4. (2, 2, 1)
Show answer and solution

Answer: (A) (2, 1, 2)

SolutionThe foot lies along the normal: (2t, t, 2t). Substituting: 4t + t + 4t = 9, so t = 1. Foot = (2, 1, 2).
Common trap(4, 2, 4) just doubles the normal vector without checking that the point lies on the plane.
CirclesEasy

Q8. What are the centre and radius of the circle x² + y² − 6x + 8y = 0?

  1. (3, −4), 5
  2. (−3, 4), 5
  3. (3, −4), 25
  4. (6, −8), 10
Show answer and solution

Answer: (A) (3, −4), 5

SolutionCompare with x² + y² + 2gx + 2fy + c = 0: g = −3, f = 4, c = 0. Centre (−g, −f) = (3, −4). Radius = √(9 + 16 − 0) = 5.
Common trapTaking the centre as (g, f) flips the signs.
EllipseMedium

Q9. What is the distance between the foci of the ellipse 9x² + 16y² = 144?

  1. √7
  2. 2√7
  3. 8
  4. 10
Show answer and solution

Answer: (B) 2√7

SolutionDividing by 144: x²/16 + y²/9 = 1, so a = 4 and b = 3. c = √(a² − b²) = √7. Distance between foci = 2c = 2√7.
Common trapGiving c (√7) instead of 2c.
VectorsEasy

Q10. What is the angle between the vectors a = î + ĵ and b = ĵ + k̂?

  1. 30°
  2. 45°
  3. 60°
  4. 90°
Show answer and solution

Answer: (C) 60°

Solutiona · b = 0 + 1 + 0 = 1, and |a| = |b| = √2. cos θ = 1/(√2 × √2) = 1/2, so θ = 60°.
Common trapAssuming they are perpendicular because they share only one component.
Straight LinesEasy

Q11. What is the acute angle between the lines y = x and y = √3 x?

  1. 15°
  2. 30°
  3. 45°
  4. 60°
Show answer and solution

Answer: (A) 15°

SolutionThe lines make angles of 45° and 60° with the x-axis, so the angle between them is 60° − 45° = 15°.
Common trapAnswering 60° by reading the slope √3 as the angle between the lines.
VectorsEasyNumerical

Q12. The vectors a = 2î − ĵ + 3k̂ and b = î + λĵ − k̂ are perpendicular. What is λ?

Numerical value question: type the answer, no options.

Show answer and solution

Answer: -1

SolutionPerpendicular vectors have a zero dot product: 2(1) + (−1)(λ) + 3(−1) = 0, so 2 − λ − 3 = 0 and λ = −1.
Common trapSign slips: −1 × λ is −λ.

Frequently asked questions

Which Coordinate Geometry & Vectors chapters are covered here?

This page covers 3D Geometry, Circles, Ellipse, Hyperbola, Parabola, Straight Lines, Vectors. Questions follow the JEE Main pattern, with multiple-choice and numerical value types.

Are these previous year JEE questions?

No. These are original JEE Main-level practice questions written on the latest pattern, with solutions and the common trap for each.