Three Dimensional Geometry Question 356
Question: A line makes the same angle $ \alpha $ with each of the x and y axes. If the angle $ \theta $ , which it makes with the z-axis, is such that $ sin^{2}\theta =2,{{\sin }^{2}}\alpha $ , then what is the value of $ \alpha $ ?
Options:
A) $ \pi /4 $
B) $ \pi /6 $
C) $ \pi /3 $
D) $ \pi /2 $
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Answer:
Correct Answer: A
Solution:
[a] Since $ l^{2}+m^{2}+n^{2}=1 $
$ \therefore {{\cos }^{2}}\alpha +{{\cos }^{2}}\alpha +{{\cos }^{2}}\theta =1 $ ????(i) ( $ \because $ A line makes the same angle $ \alpha $ with x and y-axes and $ \theta $ with z-axis) Also, $ {{\sin }^{2}}\theta =2{{\sin }^{2}}\alpha $
$ \Rightarrow 1={{\cos }^{2}}\theta =2(1-cos^{2}\alpha ) $ $ (\therefore {{\sin }^{2}}A+{{\cos }^{2}}A=1) $
$ \Rightarrow {{\cos }^{2}}\theta =2{{\cos }^{2}}\alpha -1 $ ????(ii)
$ \therefore $ From equation (i) and (ii) $ 2{{\cos }^{2}}\alpha +2{{\cos }^{2}}\alpha -1=1 $
$ \Rightarrow 4{{\cos }^{2}}\alpha =2\Rightarrow \cos \alpha =\pm \frac{1}{\sqrt{2}}\Rightarrow \alpha =\frac{\pi }{4},\frac{3\pi }{4} $
BETA
