MHT CET · Maths · Straight Lines
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 \(150^{\circ}\)
- C \(165^{\circ}\)
- D \(180^{\circ}\)
Answer & Solution
Correct Answer
(C) \(165^{\circ}\)
Step-by-step Solution
Detailed explanation
\(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1 \)
\( \Rightarrow \cos ^2 45^{\circ}+\cos ^2 \beta+\cos ^2 \beta=1 \)
\( \Rightarrow 2 \cos ^2 \beta=1-\frac{1}{2}=\frac{1}{2} \)
\( \Rightarrow \cos ^2 \beta=\frac{1}{4} \)
\( \therefore \beta=60^{\circ}=\gamma \)
\( \Rightarrow \alpha+\beta+\gamma=165^{\circ}\) \(\ldots .(\because \beta=\gamma)\)
\( \Rightarrow \cos ^2 45^{\circ}+\cos ^2 \beta+\cos ^2 \beta=1 \)
\( \Rightarrow 2 \cos ^2 \beta=1-\frac{1}{2}=\frac{1}{2} \)
\( \Rightarrow \cos ^2 \beta=\frac{1}{4} \)
\( \therefore \beta=60^{\circ}=\gamma \)
\( \Rightarrow \alpha+\beta+\gamma=165^{\circ}\) \(\ldots .(\because \beta=\gamma)\)
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