Question

If a line makes angles $\alpha ,\beta$, $\gamma$ with the coordinate axes, then

• A. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma –1=0$
• B. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma –2=0$
• C. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1=0$
• D. $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +2=0$

Answer 1

The correct option is C

$\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1=0$

Explanation for the correct option:

Given, The line makes angles $\alpha ,\beta$ and $\gamma$ with coordinate axis

$\therefore l^2+m^2+n^2=1$

$\Rightarrow$${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$ ….(1)

Now,

$\begin{array}{rcl}\cos 2\alpha +\cos 2\beta +\cos 2\gamma & =& \left(2{\cos }^2\alpha -1\right)+\left(2{\cos }^2\beta -1\right)+\left(2{\cos }^2\gamma -1\right)\\ & =& 2\left({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma \right)-3\\ & =& 2\left(1\right)-3\\ & =& -1\end{array}$

$\mathbf{\therefore }\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{2}\mathit{\alpha }\mathbf{}\mathbf{+}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{2}\mathit{\beta }\mathbf{}\mathbf{+}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathbf{2}\mathit{\gamma }\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$

Hence, Option ‘C’ is Correct.