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Let \( f(x)=\log _{e}(\sin x),(0x\pi) \) and \( g(x)=\sin ^{-1}\left(e^{-x}\right) \), \( (x \geq 0) \). If \( \alpha \) is a positive real number such that
\( \mathrm{P} \) \( a=(f o g)^{\prime}(\alpha) \) and \( b=(f o g)(\alpha) \), then (2019 Main, 10 April II)
(a) \( a \alpha^{2}-b \alpha-a=0 \)
(b) \( a \alpha^{2}-b \alpha-a=1 \)
IV
(c) \( a \alpha^{2}+b \alpha-a=-2 \alpha^{2} \)
(d) \( a \alpha^{2}+b \alpha+a=0 \)
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