A) \[a\cos \theta \sin \theta \]
B) \[ab\sin \theta \cos \theta \]
C) \[\frac{1}{2}ab\]
D) \[ab\]
Correct Answer: D
Solution :
Area\[=\frac{1}{2}\,\,\left| \begin{matrix} a\,\cos \theta & b\,\sin \theta & 1 \\ -a\,\sin \theta & b\,\cos \theta & 1 \\ -a\,\cos \theta & -b\,\sin \theta & 1 \\ \end{matrix}\, \right|\] \[=\frac{1}{2}\,(a\times b)\,\left| \,\begin{matrix} \cos \theta & \sin \theta & 1 \\ -\sin \theta & \cos \theta & 1 \\ -\cos \theta & -\sin \theta & 1 \\ \end{matrix}\, \right|\] \[=\frac{ab}{2}[\cos \theta \,(\cos \theta +\sin \theta )-\sin \theta \,(-\sin \theta +\cos \theta )\]\[+1\,({{\sin }^{2}}\theta +{{\cos }^{2}}\theta )]\] \[=\frac{ab}{2}(1+1)=ab\].
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