Question

In the figure, $ABCD$ is a unit square. $A$ circle is drawn with centre $O$ on the extended line $CD$ and passing through $A$. If the diagonal $AC$ is tangent to the circle, then the area of the shaded region is

• A. $\frac{9-\pi }{6}$
• B. $\frac{8-\pi }{6}$
• C. $\frac{7-\pi }{4}$
• D. $\frac{6-\pi }{4}$

Answer 1

Answer: (D)

Given, $ABCD$ is a square
$AB=CD=AD=BC=1$
$AC$ is tangent of circle
$\angle OAC=90^{\circ }$
$\angle CAD=45^{\circ }$
$\therefore \angle OAD=45^{\circ }$
$\therefore OA=\sqrt{2}$

$\therefore$ Area of shaded region
$=$Area of square $+$ Area of $\Delta AOD-$ Area of sector
$=1+\frac{1}{2}\times 1-\frac{45}{360}\times (\sqrt{2}{)}^2\cdot \pi$
$=1+\frac{1}{2}-\frac{\pi }{4}=\frac{3}{2}-\frac{\pi }{4}=\frac{6-\pi }{4}$