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Three Dimensional Geometry — Class 12 solved problems

37 problems from this chapter, each solved step by step.

  1. Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of the triangles ABC, ACD and ADB be 5, 6 and 7 square uni

    18 + √110 square units

  2. Let L_1: (x-1)/(1) = (y-2)/(1) = (z-1)/(2) and L_2: (x+1)/(1) = (y-2)/(2) = (z)/(4) be two lines. Let L_3 be a line passing through the point (α, β,) and be per

    (271)/(5)

  3. The line L_1 is parallel to the vector a = -3 i + 2 j + 4 k and passes through the point (7, 6, 2) and the line L_2 is parallel to the vector b = 2 i + j + 3 k

    The shortest distance between the lines L_1 and L_2 is (69)/(√342).

  4. The distance of the line (x-2)/(2)=(y-6)/(3)=(z-3)/(4) from the point (1,4,0) along the line (x)/(4)=(y-2)/(2)=(z+3)/(3) is:

    The distance is (√4221)/(8).

  5. Find the shortest distance between the skew lines: r⃗₁ = (i + j + k) + λ(2i + 3j + 4k) and r⃗₂ = (2i + 3j + 5k) + μ(i + 2j + 3k).

    The shortest distance between the skew lines is (√6)/(6) units.

  6. Find the equation of the plane passing through the point (1, 2, -3) and perpendicular to the planes x + 2y + 3z = 4 and 2x - 3y + 4z = 5.

    The equation of the plane is 17x + 2y - 7z - 42 = 0.

  7. Let the line passing through the points (-1, 2, 1) and parallel to the line (x-1)/(2) = (y+1)/(3) = (z)/(4) intersect the line (x+2)/(3) = (y-3)/(2) = (z-4)/(3)

    5√5

  8. Let the shortest distance between the lines (x-3)/(3) = (y-α)/(-1) = (z-3)/(1) and (x+3)/(-3) = (y+7)/(2) = (z-β)/(4) be 3√30. Then the positive value of 5α + β

    46

  9. Consider the lines L_1: x - 1 = y - 2 = z and L_2: x - 2 = y - 2 = z - 1. Let the feet of the perpendiculars from the point P(5,1,-3) on the lines L_1 and L_2 b

    67

  10. Let A be the point of intersection of the lines L_1: (x-7)/(1)=(y-5)/(0)=(z-3)/(-1), and L_2: (x-1)/(3)=(y+3)/(4)=(z+7)/(5). Let B and C be points on L_1 and L_

    54

  11. If the image of the point P(1, 0, 3) in the line joining the points A(4, 7, 1) and B(3, 5, 3) is Q(α, β,), then α + β + is equal to:

    (46)/(3)

  12. The distance of the point (7,10,11) from the line (x-4)/(1)=(y-4)/(0)=(z-2)/(3) along the line (x-9)/(2)=(y-13)/(-3)=(z-17)/(6) is:

    26

  13. 35. (a) Represent the equations of lines l_1 and l_2 in vector form and check whether they are intersecting or not. l_1: (x+3)/(-3) = (y-1)/(1) = (z-5)/(5) l_2:

    The vector form of line l_1 is r = (-3 i + j + 5 k) + (-3 i + j + 5 k). The vector form of line l_2 is r = (- i + 2 j + 5 k) + (- i + 2 j + 5 k). The lines inte

  14. (b) Opposite sides of a square are along the lines: r = i + 2 j - 4 k + (2 i + 3 j + 6 k) r = 3 i + 3 j - 5 k + (2 i + 3 j + 6 k) Find the area of the square if

    The area of the square is (293)/(49) square units, and the value of p is -2.

  15. If the image of the point (4, 4, 3) in the line (x - 1)/(2) = (y - 2)/(1) = (z - 1)/(3) is (α, β,), then α + β + is equal to

    9

  16. If the square of the shortest distance between the lines (x-2)/(1)=(y-1)/(2)=(z+3)/(3) and (x+1)/(2)=(y+3)/(4)=(z+5)/(5) is (m)/(n), where m, n are coprime numb

    9

  17. Let P be the image of the point Q(7, -2, 5) in the line L: (x - 1)/(2) = (y + 1)/(3) = (z)/(4) and R(5, p, q) be a point on L. Then the square of the area of PQ

    155

  18. Find the shortest distance between the lines r = (i + 2j + k) + lambda*(i - j + k) and r = (2i - j - k) + mu*(2i + j + 2k).

    The shortest distance between the given lines is (3√2)/(2) units.

  19. A line makes angles alpha, beta, gamma and delta with the four diagonals of a cube, prove that cos²(alpha) + cos²(beta) + cos²(gamma) + cos²(delta) = 4/3.

    cos²(α) + cos²(β) + cos²() + cos²() = 4/3 is proven.

  20. Let P be the foot of the perpendicular from the point (1,2,2) on the line L: (x-1)/(1)=(y+1)/(-1)=(z-2)/(2). Let the line r=(- i+ j-2 k)+ ( i- j+ k), R, interse

    27

  21. Let A(x, y, z) be a point in xy -plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1). Let B = (1, 4, -1) and C = (2, 0, -2). Then a

    Statement (S1) is true and statement (S2) is false.

  22. If the equation of the line passing through the point (0, -(1)/(2), 0) and perpendicular to the lines r = ( i + a j + b k) and r = ( i - j - 6 k) + (-b i + a j

    14

  23. The perpendicular distance, of the line (x-1)/(2)=(y+2)/(-1)=(z+3)/(2) from the point P(2,-10,1), is:

    3√5

  24. Let the line L pass through (1, 1, 1) and intersect the lines (x - 1)/(2) = (y + 1)/(3) = (z - 1)/(4) and (x - 3)/(1) = (y - 4)/(2) = (z)/(1). Then, which of th

    The point (-1, 0, -3) lies on the line L.

  25. Let the vertices Q and R of the triangle PQR lie on the line (x+3)/(5)=(y-1)/(2)=(z+4)/(3), QR = 5, and the coordinates of the point P be (0,2,3). If the area o

    (5√21)/(2)

  26. Let L_1: (x-1)/(2) = (y-2)/(3) = (z-3)/(4) and L_2: (x-2)/(3) = (y-4)/(4) = (z-5)/(6) be two lines. Then which of the following points lies on the line of the s

    The point (5, 8, 11) lies on the line of the shortest distance between L_1 and L_2.

  27. If the shortest distance between the lines (x - 1)/(2) = (y - 2)/(3) = (z - 3)/(4) and (x)/(1) = (y)/(α) = (z - 5)/(1) is (5)/(√6), then the sum of all possible

    -3

  28. Each of the angles β and that a given line makes with the positive y - and z -axes, respectively, is half of the angle that this line makes with the positive x

    The sum of all possible values of the angle β is (3π)/(4).

  29. Let a line pass through two distinct points P(-2,-1,3) and Q, and be parallel to the vector 3 i+2 j+2 k. If the distance of the point Q from the point R(1,3,3)

    34

  30. Let P be the foot of the perpendicular from the point Q(10, -3, -1) on the line (x - 3)/(7) = (y - 2)/(-1) = (z + 2)/(-2). Then the area of the right-angled tri

    The area of the right-angled triangle PQR is (1)/(2)√1346 square units.

  31. Let L_1: (x-1)/(3) = (y-1)/(4) = (z+1)/(0) and L_2: (x-2)/(2) = (y)/(0) = (z+4)/(0), α R, be two lines, which intersect at the point B. If P is the foot of perp

    175.5

  32. MODIFIED: Find the shortest distance between the skew lines: r⃗₁ = (i + j + k) + λ(2i + 3j + 4k) and r⃗₂ = (2i + 3j + 5k) + μ(i + 2j + 3k).

    The shortest distance between the given skew lines is (√6)/(6) units.

  33. Let a straight line L pass through the point P(2,-1,3) and be perpendicular to the lines (x-1)/(4)=(y+1)/(4)=(z-3)/(2) and (x-3)/(4)=(y-2)/(3)=(z+2)/(4). If the

    The distance between the points P and Q is (6√5)/(5).

  34. Let a line passing through the point (4,1,0) intersect the line L_1: (x-1)/(2)=(y-2)/(3)=(z-3)/(4) at the point A(α,β,) and the line L_2: x-6 = y - z + 4 at the

    2

  35. Let the area of the triangle formed by the lines x + 2 = y - 1 = z, (x - 3)/(5) = (y)/(-1) = (z - 1)/(1), and (x)/(-3) = (y - 3)/(3) = (z - 2)/(1) be A. Then A²

    56

  36. Let in a ABC, the length of the side AC be 6, the vertex B be (1,2,3) and the vertices A, C lie on the line (x-6)/(3) = (y-7)/(2) = (z-7)/(2). Then the area (in

    The area of ABC is (3√561)/(17) square units.

  37. Let the values of for which the shortest distance between the lines (x-1)/(2) = (y-2)/(3) = (z-3)/(4) and (x -)/(3) = (y - 4)/(4) = (z - 5)/(5) is (1)/(√6) be _

    (5√2)/(3)

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