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MCQMediumJEE 2024Direction Cosines & Ratios

JEE Mathematics 2024 Question with Solution

The distance of point P from the x-axis, given the conditions:

OP = γ\gamma; the angle between OQ and the positive x-axis is θ\theta; and the angle between OP and the positive z-axis is ϕ\phi.

  • A

    γ(1−sin⁡2(ϕ)cos⁡2(θ))\gamma\sqrt{(1 - \sin^2(\phi) \cos^2(\theta))}

  • B

    γ(1+cos⁡2(θ)sin⁡2(ϕ))\gamma\sqrt{(1 + \cos^2(\theta) \sin^2(\phi))}

  • C

    γ(1−sin⁡2(θ)cos⁡2(ϕ))\gamma\sqrt{(1 - \sin^2(\theta) \cos^2(\phi))}

  • D

    γ(1+cos⁡2(ϕ)sin⁡2(θ))\gamma\sqrt{(1 + \cos^2(\phi) \sin^2(\theta))}

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: OP = γ\gamma, the angle between OQ and the positive x-axis is θ\theta, and the angle between OP and the positive z-axis is ϕ\phi.

Find: The distance of point P from the x-axis.

The solution concludes that the correct option is A. However, the displayed working in the solution is unrelated to this question and discusses common chords of two circles. Hence, no valid derivation for this question could be extracted from the solution.

Using the answer indicated on the solution's, the correct option is A, which is

γ1−sin⁡2(ϕ)cos⁡2(θ)\gamma\sqrt{1 - \sin^2(\phi)\cos^2(\theta)}

Common mistakes

  • Using the unrelated circle-based working from the solution for this 3D geometry question is incorrect because it does not correspond to the given variables γ\gamma, θ\theta, and ϕ\phi. Always ensure the solution method matches the actual question statement.

  • Confusing the distance from the x-axis with the x-coordinate of point P is incorrect. In three dimensions, distance from the x-axis depends on the perpendicular separation from that axis, not merely one coordinate.

  • Interchanging the roles of angles θ\theta and ϕ\phi can lead to an incorrect trigonometric expression. First identify clearly which angle is measured from the x-axis and which is measured from the z-axis.

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