Question

A line makes the same angle $\theta$ with each of the $x$ and $z-$axes. If the angle $\beta ,$ which it makes with the $y-$axis, is such that ${\sin }^2\beta =3{\sin }^2\theta ,$ then ${\cos }^2\theta =$

• A. $\frac{2}{3}$
• B. $\frac{1}{5}$
• C. $\frac{3}{5}$
• D. $\frac{2}{5}$

Answer 1

The correct option is C $\frac{3}{5}$

Let $(l,m,n)$ are d.c's of the line.
So $l=\cos \theta ,m=\cos \beta ,$ and $n=\cos \theta$
We know that $l^2+m^2+n^2=1$
So ${\cos }^2\theta +{\cos }^2\beta +{\cos }^2\theta =1\Rightarrow 2{\cos }^2\theta ={\sin }^2\beta$
Hence, from the given relation, we have:
$\Rightarrow 2{\cos }^2\theta =3{\sin }^2\theta \Rightarrow 2{\cos }^2\theta =3(1-{\cos }^2\theta )$
$\Rightarrow {\cos }^2\theta =\frac{3}{5}$