Question

Let $\alpha ,\beta$ be two distinct roots of a $\cos \theta +b\sin \theta =c$, where $a,b$ and $c$ are three real constants and $\theta ϵ[0,2\pi ]$. Then $\alpha +\beta$ is also a root of the same equation if

• A. $a+b=c$
• B. $b+c=a$
• C. $c+a=b$
• D. $c=a$

Answer 1

The correct option is D $c=a$

$a\left(\frac{1-{\tan }^2\frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}\right)+\frac{2b\tan \frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}=c$
$\Rightarrow (c+a){\tan }^2\frac{\theta }{2}-2b\tan \frac{\theta }{2}+(c-a)=0$
so, $\tan \frac{\alpha +\beta }{2}=\frac{\frac{2b}{c+a}}{1-\frac{c-a}{c+a}}=\frac{b}{a}$
$\frac{b}{a}$ is a root of the equation,
$(c+a)\frac{b^2}{a^2}-2b\left(\frac{b}{a}\right)+c-a=0$
so, $c=a$