Three Dimensional Geometry Question 284
Question: A line makes the same angle $ \theta $ , with each of the x and z?axis. If the angle $ \beta $ , which it makes with y-axis is such that $ {{\sin }^{2}}\beta =3{{\sin }^{2}}\theta , $ then $ {{\cos }^{2}}\theta $ equals
[AIEEE 2004]
Options:
A) $ \frac{3}{5} $
B) $ \frac{2}{3} $
C) $ \frac{1}{5} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Here, $ l=\cos \theta ,,m=\cos \beta ,,n=\cos \theta $ , $ (\because l=n) $ Now, $ l^{2}+m^{2}+n^{2}=1\Rightarrow 2{{\cos }^{2}}\theta +{{\cos }^{2}}\beta =1 $
$ \Rightarrow $ $ 2{{\cos }^{2}}\theta ={{\sin }^{2}}\beta $ Given, $ {{\sin }^{2}}\beta =3{{\sin }^{2}}\theta $
Þ $ 2{{\cos }^{2}}\theta =3{{\sin }^{2}}\theta $
Þ $ 5{{\cos }^{2}}\theta =3,,\therefore ,{{\cos }^{2}}\theta =\frac{3}{5}. $
BETA
