If $α, β$ are different values of x satisfying the equation a cos x +b sina x = c, where a, b and c are constants, then $\tan (\frac{α + β}{2})$ is

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a+b

b-a

a/b

b/a

answer is D.

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Detailed Solution

Given $α, β$ are different values of x satisfying the
equation a cos x + b sin x = c,
a cos $α$ +b sin $α$ =c -----(i)
a cos $β$ +b sin $β$ = c -------(ii)
From (i) and (ii)
$\begin{array}{l} a \cos α + b \sin α = a \cos β + b \sin β \\ \Rightarrow a (\cos α - \cos β) + b (\sin α - \sin β) = 0 \\ \Rightarrow a (- 2 \sin \frac{α + β}{2} \cdot \sin \frac{α - β}{2}) + b (2 \cos \frac{α + β}{2} \cdot \sin \frac{α - β}{2}) \\ = 0 \\ \Rightarrow 2 \sin (\frac{α - β}{2}) [b \cos \frac{α + β}{2} - a \sin \frac{α + β}{2}] = 0 \\ \Rightarrow \sin \frac{α - β}{2} \neq 0 [∴ α \neq β] \\ ∴ b \cos \frac{α + β}{2} - a \sin \frac{α + β}{2} = 0 ∴ \tan \frac{α + β}{2} = \frac{b}{a} \end{array}$