A line makes the angle $\theta$ with each of the x and z axes. If the angle $\beta$ which it makes with y-axis is such that $\sin ^2 \beta=3 \sin ^2 \theta$, then the value of $\cos ^2 \theta$ is
Answer & explanation
Correct answer: option 3
We know that sum of squares of direction ratios is 1
so $\cos ^2 \theta+\cos ^2 \theta+\cos ^2 \beta=1$
$2 \cos ^2 \theta+1-\sin ^2 \beta=1$
so $2 \cos ^2 \theta =\sin ^2 \beta$ (given $\sin^2 \beta = 3 \sin^2 \theta$)
$2 \cos ^2 \theta =3 \sin ^2 \theta$
so $2 \cos ^2 \theta =3-3 \cos ^2 \theta$
$\Rightarrow \cos ^2 \theta =\frac{3}{5}$
Option: 3