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Three Dimensional Geometry · Question 35

JEE Main 2025 (Online) 7th April Morning ShiftMCQMedium📈 Topic in 35% papers+4−1
1

Let the line L pass through (1, 1, 1) and intersect the lines x-12 = y+13 = z-14 and x-31 = y-42 = z1. Then, which of the following points lies on the line L?

  1. (7, 15, 13)
  2. (4, 22, 7)
  3. (10, -29, -50)
  4. (5, 4, 3)

Answer

(1) (7, 15, 13)

Solution

Let P(1, 1, 1). Line L1: through A(1, -1, 1), direction (2, 3, 4); line L2: through B(3, 4, 0), direction (1, 2, 1).

Since L passes through P and meets L1, L lies in the plane containing P and L1.

AP = (0, 2, 0); normal = (2, 3, 4) (0, 2, 0) = (-8, 0, 4) (2, 0, -1)

plane 1: 2(x-1) - (z-1) = 0, i.e. 2x - z = 1

Similarly L lies in the plane containing P and L2:

BP = (-2, -3, 1); normal = (1, 2, 1) (-2, -3, 1) = (5, -3, 1)

plane 2: 5(x-1) - 3(y-1) + (z-1) = 0, i.e. 5x - 3y + z = 3

direction of L = (2, 0, -1) (5, -3, 1) = (-3, -7, -6) (3, 7, 6)

L: x-13 = y-17 = z-16 = , i.e. (1+3 , 1+7 , 1+6 )

= 2 gives (7, 15, 13), which lies on L.

Check: 2(7) - 13 = 1 and 5(7) - 3(15) + 13 = 3 ✓. The other points do not satisfy both plane equations.

Answer: (1)

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2

If the shortest distance between the lines x-12 = y-23 = z-34 and x1 = y = z-51 is 5 6, then the sum of all possible values of is

  1. 32
  2. 3
  3. -3
  4. - 32
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3

If the equation of the line passing through the point (0, - 12, 0 ) and perpendicular to the lines r = ( i + a j + b k ) and r = ( i - j - 6 k ) + (-b i + a j + 5 k ) is x-1-2 = y+4d = z-c-4, then a + b + c + d is equal to :

  1. 13
  2. 14
  3. 12
  4. 10
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4

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  2. 147
  3. 139
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5

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  2. 40
  3. 48
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6

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7

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  1. 63
  2. 57
  3. 60
  4. 54
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8

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  3. 8
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Line L1 passes through the point (1, 2, 3) and is parallel to z-axis. Line L2 passes through the point ( , 5, 6) and is parallel to y-axis. Let for = 1, 2, 2 < 1, the shortest distance between the two lines be 3. Then the square of the distance of the point ( 1, 2, 7) from the line L1 is

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  2. 32
  3. 40
  4. 37
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