Q. If $\sin{\theta + \cos{\theta = h}}$, then the quadratic equation having $\sin\theta$ and $\cos\theta$ as its roots, is
A) $x^{2} - hx + \left( h^{2} - 1 \right) = 0$
B) $2x^{2} - 2hx + \left( h^{2} - 1 \right) = 0$
C) $x^{2} - hx + 2\left( h^{2} - 1 \right) = 0$
D) $x^{2} - 2hx + \left( h^{2} - 1 \right) = 0$
Explanation
(b)
Given, $\sin{\theta + \cos{\theta = h}}$
$\Rightarrow \sin^{2}{\theta + \cos^{2}{\theta + 2\sin{\theta\cos{\theta = h^{2}}}}}$ [squaring]
$\Rightarrow \sin{\theta\cos{\theta = \frac{h^{2} - 1}{2}}}$
The quadratic equation having the roots $\sin\theta$ and $\cos\theta$ is
$x^{2} - (\sin{\theta + \cos{\theta)x + \sin{\theta\cos{\theta = 0}}}}$
$\therefore\ \ \ \ 2x^{2} - 2hx + \left( h^{2} - 1 \right) = 0$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.