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JEE Main Three Dimensional Geometry questions

67 Mathematics questions on Three Dimensional Geometry as asked in JEE Main, newest first. 33 come with a worked solution.

  1. The distance of the point P (4, 6, –2) from the line passing through the point (–3, 2, 3) and parallel to a line with direction ratios 3, 3, –1 is equal to :

    2023 · with solution

  2. The shortest distance between the lines x + 1 = 2y = -12z and x = y + 2 = 6z - 6 is

    2023

  3. The distance of the point (7, -3, -4) from the plane passing through the points (2, -3, 1), (-1, 1, -2) and (3, -4, 2) is:

    2023 · with solution

  4. Let the plane containing the line of intersection of the planes P1 : x + (λ + 4)y + z = 1 and P2 : 2x + y + z = 2 pass through the points (0, 1, 0) and (1, 0, 1). Then the distance of the point (2λ, λ, -λ) from the…

    2023 · with solution

  5. The distance of the point (-1, 9, -16) from the plane 2x + 3y - z = 5 measured parallel to the line (x+4)/3 = (2-y)/4 = (z-3)/12 is

    2023 · with solution

  6. The foot of perpendicular of the point (2, 0, 5) on the line (x+1)/2 = (y-1)/5 = (z+1)/(-1) is (α, β, γ). Then, which of the following is NOT correct?

    2023 · with solution

  7. Consider the lines L_1 and L_2 given by L_1 : (x-1)/2 = (y-3)/1 = (z-2)/2 L_2 : (x-2)/1 = (y-2)/2 = (z-3)/3 A line L_3 having direction ratios 1, -1, -2, intersects L_1 and L_2 at the points P and Q respectively. Then…

    2023

  8. If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel to the planes x + 2y + z = 0 and 3y - z = 3 is (α, β, γ), then α + β + γ is equal to

    2023 · with solution

  9. If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line (x-1)/2 = (y+1)/(-1) = (z-2)/0 is a, then 28a^2 is equal to ______.

    2023 · with solution

  10. If the shortest between the lines fracx+√(6)2 = fracy-√(6)3 = fracz-√(6)4 and (x-λ)/3 = fracy-2√(6)4 = fracz+2√(6)5 is 6, then the square of sum of all possible values of λ is

    2023 · with solution

  11. Let the equation of the plane passing through the line x - 2y - z - 5 = 0 = x + y + 3z - 5 and parallel to the line x + y + 2z - 7 = 0 = 2x + 3y + z - 2 be ax + by + cz = 65. Then the distance of the point (a, b, c)…

    2023

  12. Let the co-ordinates of one vertex of triangle ABC be A(0, 2, a) and the other two vertices lie on the line (x+a)/5 = (y-1)/2 = (z+4)/3. For a ∈ Z, if the area of triangle ABC is 21 sq. units and the line segment BC has…

    2023

  13. Let the equation of the plane P containing the line x + 10 = (8-y)/2 = z be ax + by + 3z = 2(a+b) and the distance of the plane P from the point (1, 27, 7) be c. Then a^2 + b^2 + c^2 is equal to _______.

    2023 · with solution

  14. The distance of the point having position vector -i + 2j + 6k from the straight line passing through the point (2, 3, -4) and parallel to the vector, 6i + 3j - 4k is

    2019

  15. The vertices B and C of a triangle ABC lie on the line, (x+2)/3 = (y-1)/0 = z/4 such that BC = 5 units. Then, the area (in sq units) of this triangle, given that the point A(1, -1, 2) is

    2019 · with solution

  16. The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines r = (i + j) + λ(i + 2j - k) and r = (i + j) + μ(-i + j - 2k) is

    2019 · with solution

  17. Let √(3)i + j, i + √(3)j and βi + (1-β)j respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is…

    2019 · with solution

  18. If the length of the perpendicular from the point (β, 0, β) (β ≠ 0) to the line, x/1 = (y-1)/0 = (z+1)/(-1) is √(3/2), then β is equal to

    2019

  19. A plane which bisects the angle between the two given planes 2x - y + 2z - 4 = 0 and x + 2y + 2z - 2 = 0, passes through the point

    2019

  20. If the line (x-2)/3 = (y+1)/2 = (z-1)/(-1) intersects the plane 2x + 3y - z + 13 = 0 at a point P and the plane 3x + y + 4z = 16 at a point Q, then PQ is equal to

    2019 · with solution

  21. A perpendicular is drawn from a point on the line (x-1)/2 = (y+1)/(-1) = z/1 to the plane x + y + z = 3 such that the foot of the perpendicular Q also lies on the plane x - y + z = 3. Then, the coordinates of Q are

    2019 · with solution

  22. If the plane 2x - y + 2z + 3 = 0 has the distances 1/3 and 2/3 units from the planes 4x - 2y + 4z + λ = 0 and 2x - y + 2z + μ = 0, respectively, then the maximum value of λ + μ is equal to

    2019

  23. A tetrahedron has vertices P(1,2,1), Q(2,1,3), R(-1,1,2), O(0,0,0). The angle between the faces OPQ and PQR is

    2019 · with solution

  24. If Q(0, -1, -3) is the image of the point P in the plane 3x - y + 4z = 2 and R is the point (3, -1, -2), then the area (in sq units) of triangle PQR is

    2019

  25. If a point R(4, y, z) lies on the line segment joining the points P(2, -3, 4) and Q(8, 0, 10), then the distance of R from the origin is

    2019 · with solution

  26. Let P be the plane, which contains the line of intersection of the planes, x + y + z - 6 = 0 and 2x + 3y + z + 5 = 0 and it is perpendicular to the XY-plane. Then, the distance of the point (0, 0, 256) from P is equal to

    2019

  27. A plane passing through the points (0, -1, 0) and (0, 0, 1) and making an angle (π)/4 with the plane y - z + 5 = 0, also passes through the point

    2019

  28. If the line (x-1)/2 = (y+1)/3 = (z-2)/4 meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is

    2019

  29. The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5, which is perpendicular to the plane x - y + z = 0 is

    2019 · with solution

  30. The length of the perpendicular from the point (2, -1, 4) on the straight line, (x+3)/10 = (y-2)/(-7) = z/1 is

    2019

  31. The magnitude of the projection of the vector 2i + 3j + k on the vector perpendicular to the plane containing the vectors i + j + k and i + 2j + 3k, is

    2019

  32. The equation of a plane containing the line of intersection of the planes 2x - y - 4 = 0 and y + 2z - 4 = 0 and passing through the point (1, 1, 0) is

    2019

  33. If an angle between the line, (x+1)/2 = (y-2)/1 = (z-3)/(-2) and the plane, x - 2y - kz = 3 is cos^-1 (frac2√(2)3 ), then value of k is

    2019

  34. Let S be the set of all real values of λ such that a plane passing through the points (-λ^2, 1, 1), (1, -λ^2, 1), (1, 1, -λ^2) also passes through the point (-1, -1, 1). Then S is equal to

    2019

  35. The perpendicular distance from the origin to the plane containing the two lines, (x+2)/3 = (y-2)/5 = (z+5)/7 and (x-1)/1 = (y-4)/4 = (z+7)/7, is

    2019

  36. Two lines (x-3)/1 = (y+1)/3 = (z-6)/(-1) and (x+5)/7 = (y-2)/(-6) = (z-3)/4 intersect at the point R. The reflection of R in the xy-plane has coordinates

    2019 · with solution

  37. If the point (2, α, β) lies on the plane which passes through the points (3, 4, 2) and (7, 0, 6) and is perpendicular to the plane 2x - 5y = 15, then 2α - 3β is equal to

    2019

  38. The plane containing the line (x-3)/2 = (y+2)/(-1) = (z-1)/3 and also containing its projection on the plane 2x + 3y - z = 5, contains which one of the following points?

    2019 · with solution

  39. The direction ratios of normal to the plane through the points (0, -1, 0) and (0, 0, 1) and making an angle π/4 with the plane y - z + 5 = 0 are

    2019

  40. The plane which bisects the line segment joining the points (-3, -3, 4) and (3, 7, 6) at right angles, passes through which one of the following points?

    2019

  41. The plane passing through the point (4, -1, 2) and parallel to the lines (x+2)/3 = (y-2)/(-1) = (z+1)/2 and (x-2)/1 = (y-3)/2 = (z-4)/3 also passes through the point

    2019 · with solution

  42. Let A be a point on the line r = (1 - 3μ)i + (μ - 1)j + (2 + 5μ)k and B(3, 2, 6) be a point in the space. Then, the value of μ for which the vector overrightarrowAB is parallel to the plane x - 4y + 3z = 1 is

    2019

  43. The equation of the plane containing the straight line x/2 = y/3 = z/4 and perpendicular to the plane containing the straight lines x/3 = y/4 = z/2 and x/4 = y/2 = z/3 is

    2019

  44. The plane through the intersection of the planes x + y + z = 1 and 2x + 3y - z + 4 = 0 and parallel to Y-axis also passes through the point

    2019 · with solution

  45. If L_1 is the line of intersection of the planes 2x - 2y + 3z - 2 = 0, x - y + z + 1 = 0 and L_2 is the line of intersection of the planes x + 2y - z - 3 = 0, 3x - y + 2z - 1 = 0, then the distance of the origin from…

    2018

  46. The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x + y + z = 7 is

    2018 · with solution

  47. If the image of the point P(1, -2, 3) in the plane 2x + 3y - 4z + 22 = 0 measured parallel to the line x/1 = y/4 = z/5 is Q, then PQ is equal to

    2017 · with solution

  48. The distance of the point (1, 3, -7) from the plane passing through the point (1, -1, -1) having normal perpendicular to both the lines (x-1)/1 = (y+2)/(-2) = (z-4)/3 and (x-2)/2 = (y+1)/(-1) = (z+7)/(-1), is

    2017

  49. The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y - 2z = 5 and 3x - 6y - 2z = 7 is

    2017 · with solution

  50. The distance of the point (1, -5, 9) from the plane x - y + z = 5 measured along the line x = y = z is

    2016 · with solution

  51. If the line, (x-3)/2 = (y+2)/(-1) = (z+4)/3 lies in the plane, lx + my - z = 9, then l^2 + m^2 is equal to

    2016 · with solution

  52. Let P be the image of the point (3, 1, 7) with respect to the plane x - y + z = 3. Then, the equation of the plane passing through P and containing the straight line x/1 = y/2 = z/1 is

    2016

  53. The equation of the plane containing the lines 2x - 5y + z = 3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z = 1 is

    2015 · with solution

  54. The distance of the point (1, 0, 2) from the point of intersection of the line (x-2)/3 = (y+1)/4 = (z-2)/12 and the plane x - y + z = 16 is

    2015 · with solution

  55. The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l^2 = m^2 + n^2, is

    2014

  56. The image of the line (x-1)/3 = (y-3)/1 = (z-4)/(-5) in the plane 2x - y + z + 3 = 0 is the line

    2014

  57. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is

    2013 · with solution

  58. If the lines (x-2)/1 = (y-3)/1 = (z-4)/k and (x-1)/k = (y-4)/2 = (z-5)/1 are coplanar, then k can have

    2012 · with solution

  59. The equation of a plane passing through the line of intersection of the planes x + 2y + 3z = 2 and x - y + z = 3 and at a distance 2√(3) from the point (3, 1, -1) is

    2012

  60. The point P is the intersection of the straight line joining the points Q(2, 3, 5) and R(1, -1, 4) with the plane 5x - 4y - z = 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the…

    2012

  61. If the distance of the point P(1, -2, 1) from the plane x + 2y - 2z = α, where α > 0, is 5, then the foot of the perpendicular from P to the plane is

    2010

  62. A line with positive direction cosines passes through the point P(2, -1, 2) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals

    2009

  63. If P is (3, 2, 6) is a point in space and Q be a point on the line r = (i - j + 2k) + μ(-3i + j + 5k). Then, the value of μ for which the vector overrightarrowPQ is parallel to the plane x - 4y + 3z = 1, is

    2009 · with solution

  64. A plane passes through the point (1, -2, 1) and is perpendicular to two planes 2x - 2y + z = 0 and x - y + 2z = 4, then the distance of the plane from the point (1, 2, 2) is

    2006 · with solution

  65. A variable plane x/a + y/b + z/c = 1 at a unit distance from origin cuts the coordinate axes at A, B and C. Centroid (x, y, z) satisfies the equation 1/(x^2) + 1/(y^2) + 1/(z^2) = K. The value of K is

    2005

  66. If the lines (x-1)/2 = (y+1)/3 = (z-1)/4 and (x-3)/1 = (y-k)/2 = z/1 intersect, then the value of k is

    2004 · with solution

  67. The value of k such that (x-4)/1 = (y-2)/1 = (z-k)/2 lies in the plane 2x - 4y + z = 7, is

    2003

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