CAT 2016 Quantitative Ability

Question : 14 of 34

Marks: +1, -0

Two circles of radii 'r’ units and ‘2r‘ units intersect each other in such a way that their common chord is of the maximum possible length. What is the area (in square units) of the region that is common to the two circles?

${πr^2}/2$
${7πr^2}/6 - √3r^2$
${11πr^2}/6 - {√3r^2}/2$
${7πr^2}/6 - {√3r^2}/2$

Validate

Solution:

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)

=\;\;{πr^{2}}/{2}+\{\;\;{π(2r)^{2}}/{6}-\;\;{√{3}(2r)^{2}}/{4}\}

=\;\;{7πr^{2}}/{6}-√{3}r^{2}

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