Q76 / 150 · MHT CET 2024 (15 May Shift 2) · Mathematics

A line makes $45^{\circ}$ angle with positive X -axis and makes equal angles with positive Y -axis ad Z -axis respectively, then the sum of the three angles which the line makes with positive X -axis, Y -axis and Z -axis is

  • A
    $135^{\circ}$
  • B
    $150^{\circ}$
  • Correct answer
    C
    $165^{\circ}$
  • D
    $180^{\circ}$

Solution

$\begin{aligned} & \quad \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \\ & \Rightarrow \cos ^2 45^{\circ}+\cos ^2 \beta+\cos ^2 \beta=1 \\ & \Rightarrow 2 \cos ^2 \beta=1-\frac{1}{2}=\frac{1}{2} \\ & \Rightarrow \cos ^2 \beta=\frac{1}{4} \\ & \therefore \quad \beta=60^{\circ}=\gamma \\ & \quad \Rightarrow \alpha+\beta+\gamma=165^{\circ}\end{aligned}$ $\ldots .(\because \beta=\gamma)$

Hence, the correct answer is option: 'C'.

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