MHT CET · Maths · Three Dimensional Geometry
A line makes angles \(\frac{\alpha}{2}, \frac{\beta}{2}, \frac{Y}{2}\) with positive directions of the co-ordinate axes \((\mathrm{x}, \mathrm{y}, \mathrm{z}\) axes respectively), then \(\cos \alpha+\cos \beta+\cos \gamma\) has the value
- A 1
- B 2
- C 3
- D \(-1\)
Answer & Solution
Correct Answer
(D) \(-1\)
Step-by-step Solution
Detailed explanation
\(\cos \alpha+\cos \beta+\cos \gamma=2 \cos ^2 \frac{\alpha}{2}-1+2 \cos ^2 \frac{\beta}{2}-1+2\) \(\cos ^2 \frac{\gamma}{2}-1 \)
\( =2\left(\cos ^2 \frac{\alpha}{2}+\cos ^2 \frac{\beta}{2}+\cos ^2 \frac{\gamma}{2}\right)-3 \)
\( =2 \times 1-3\left[\because \cos ^2 \frac{\alpha}{2}+\cos ^2 \frac{\beta}{2}+\cos ^2 \frac{\gamma}{2}=1\right] \)
\( =-1\)
\( =2\left(\cos ^2 \frac{\alpha}{2}+\cos ^2 \frac{\beta}{2}+\cos ^2 \frac{\gamma}{2}\right)-3 \)
\( =2 \times 1-3\left[\because \cos ^2 \frac{\alpha}{2}+\cos ^2 \frac{\beta}{2}+\cos ^2 \frac{\gamma}{2}=1\right] \)
\( =-1\)
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