Question

For $\alpha ,\beta ,\gamma ,\theta ϵR$. Let
$A_{\theta }(\alpha ,\beta ,\gamma )=\left|\begin{array}{ccc}\cos (\alpha +\theta )& \sin (\alpha +\theta )& 1\\ \cos (\beta +\theta )& \sin (\beta +\theta )& 1\\ \cos (\gamma +\theta )& \sin (\gamma +\theta )& 1\end{array}\right|$

If $\alpha ,\beta ,\gamma$ are fixed. then $y=A_x(\alpha ,\beta ,\gamma )$ represents

• A. a straight line parallel to the x-axis
• B. a straight line through the origin
• C. a parabola with vertex at
• D. none of these

Answer 1

The correct option is A a straight line parallel to the x-axis

$A_{\theta }(\alpha ,\beta ,\gamma )=\left|\begin{array}{ccc}\cos (\alpha +\theta )& \sin (\alpha +\theta )& 1\\ \cos (\beta +\theta )& \sin (\beta +\theta )& 1\\ \cos (\gamma +\theta )& \sin (\gamma +\theta )& 1\end{array}\right|$
$=\cos (\alpha +\theta )(\sin (\beta +\theta )-\sin (\gamma +\theta ))-\sin (\alpha +\theta )(\cos (\beta +\theta )-\cos (\gamma +\theta ))+(\cos (\beta +\theta )\sin (\gamma +\theta )-\sin (\beta +\theta )\cos (\gamma +\theta ))$

$=\sin (\beta -\gamma )+\sin (\alpha -\gamma )+\sin (\gamma -\beta )$

$\therefore A_{\theta }(\alpha ,\beta ,\gamma )=\sin (\beta -\gamma )+\sin (\alpha -\gamma )+\sin (\gamma -\beta )$

Similarly, $A_x(\alpha ,\beta ,\gamma )$ is independent of $x$.

Given that, $\alpha ,\beta ,\gamma$ are fixed.

$\therefore y=A_x(\alpha ,\beta ,\gamma )=k$ where $k=\sin (\beta -\gamma )+\sin (\alpha -\gamma )+\sin (\gamma -\beta )$

A straight line parallel to the $x-axis.$

Hence, option A.