CAT 2016 Quantitative Ability

The common chord will be of the maximum length ifit is the diameter of the smaller circle.
In ΔPO’Q, PQ = 2r and PO’ = QO’ = 2r,
therefore, ΔPO’Q is an equilateral triangle.
Required area (in sq. units) of the shaded region
= 1/2 x Area of smaller circle+Area of segment PRQ
= 1/2 x Area of smaller circle + (Area of sector O’PQ - Area of triangle PO’Q)
=‌‌

πr2

2

+{‌‌

π(2r)2

6

−‌‌

√3(2r)2

4

}
=‌‌

7πr2

6

−√3r2