A) \[\frac{{{a}^{2}}}{2}(3-\pi )\]
B) \[\frac{{{a}^{2}}}{2}\left( \frac{\pi }{2}-1 \right)\]
C) \[2{{a}^{2}}(\pi -1)\]
D) \[\frac{{{a}^{2}}}{2}\left( \frac{\pi }{2}-1 \right)\]
Correct Answer: D
Solution :
Here O is the centre of the circle. From right angled \[\Delta \,\,BAC,\] \[B{{C}^{2}}=A{{C}^{2}}+A{{B}^{2}}=2{{a}^{2}}\] \[\therefore \] \[BC=\sqrt{2a}\] \[\therefore \] Radius of circle, \[OB=\frac{1}{2}\sqrt{2a}=\frac{1}{\sqrt{2}}a\] \[\therefore \] Area of semi-circle \[=\frac{1}{2}\pi {{(OB)}^{2}}\] \[=\frac{1}{2}\pi {{\left( \frac{1}{\sqrt{2}\alpha } \right)}^{2}}=\frac{1}{4}\pi {{a}^{2}}\] Area of \[\Delta \,BAC=\frac{1}{2}\times a\times a=\frac{{{a}^{2}}}{2}\] Hence, area of shaded region \[=\frac{1}{4}\pi \,{{a}^{2}}-\frac{{{a}^{2}}}{2}\] \[=\frac{{{a}^{2}}}{2}\left( \frac{\pi }{2}-1 \right)\]You need to login to perform this action.
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