Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let the direction cosines of the line be $l,m,n$.

Given that the line makes an angle $\alpha$ with the $X$ and $Z$ axes, and an angle $\beta$ with the $Y$-axis, we have:

$l=\cos \alpha$, $m=\cos \beta$, $n=\cos \alpha$

Using the property of direction cosines, $l^2+m^2+n^2=1$:

${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\alpha =1$

$2{\cos }^2\alpha +{\cos }^2\beta =1$

We are given that ${\sin }^2\beta =2{\sin }^2\alpha$. Expressing this in terms of cosine:

$1-{\cos }^2\beta =2(1-{\cos }^2\alpha )$

${\cos }^2\beta =2{\cos }^2\alpha -1$

Substituting this value of ${\cos }^2\beta$ into the first equation:

$2{\cos }^2\alpha +(2{\cos }^2\alpha -1)=1$

$4{\cos }^2\alpha =2$

${\cos }^2\alpha =\frac{1}{2}$

Answer: $\frac{1}{2}$