Question

If $x_1$ and $x_2$ are two distinct roots of the equation $a\cos x+b\sin x=c$, then $\tan \frac{x_1+x_2}{2}$ is equal to

• A. $a/b$
• B. $b/a$
• C. $c/a$
• D. $a/c$

Answer 1

The correct option is B $b/a$

$a\cos x+b\sin x=c$

$\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}+b\frac{2\tan \frac{x}{2}}{1+{\tan }^2\frac{x}{2}}=c$

$(c+a){\tan }^2\frac{x}{2}-2b\tan \frac{x}{2}+(c-a)=0$

Since, $x_1$ and $x_2$ are roots,

$\tan \frac{x_1}{2}+\tan \frac{x_2}{2}=\frac{2b}{c+a}$

and $\tan \frac{x_1}{2}.\tan \frac{x_2}{2}=\frac{c-a}{c+a}$

Hence,

$\tan \left(\frac{x_1+x_2}{2}\right)=\frac{\tan \frac{x_1}{2}+\tan \frac{x_2}{2}}{1-\tan \frac{x_1}{2}\tan \frac{x_2}{2}}$

$=\frac{\frac{2b}{c+a}}{1-\frac{c-a}{c+a}}=\frac{b}{a}$

Hence, option (b) is correct.