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MHT CET202523 Apr 2025Evening ShiftMathematicsThree Dimensional GeometryActual

The angle between the line x= y -1 2 = z -3 and the plane x+2 y+3 z=6 is ⁻¹ 5 14 , then the value of is

Options

  1. A2 3
  2. B4 3
  3. C1 3
  4. D5 3

Correct answer

A. 2 3

Step-by-step solution

Direction vector of the line: b = 1, 2, Normal vector of the plane: n = 1, 2, 3 The angle between the line and plane satisfies = | b n | | b | | n | Given = ⁻¹ 5 14 , then = 1 - 5 14 = 3 14 Compute b n = 1 + 4 + 3 = 5 + 3 Magnitudes: | b | = 5 + ^2 , | n | = 14 Substitute into the sine formula: 3 14 = |5 + 3 | 5 + ^2 14 Simplify: 3 = |5 + 3 | 5 + ^2 Squaring both sides: 9 = (5 + 3 )^2 5 + ^2 Cross-multiply and simplify: 9(5 + ^2) = 25 + 30 + 9 ^2 45 + 9 ^2 = 25 + 30 + 9 ^2 20 = 30 Solving gives = 2 3 Final answer:

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