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CAT 2008 — QA Question 19

2 CirclesEasy
Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Two circles, both of radii 1 cm, intersect such that the circumference of each one passes through the centre of the circle of the other. What is the area (in sq cm) of the intersecting region?

Answer & solution

  • A

    π3-34

  • B

    2π3+32

  • C

    4π3-32

  • D

    4π3+32

  • 2π3-32

Solution

Let O and P be the centres of the circles.

OR = OP = PR = 1cm

∴ ∆PRO is an equilateral triangle.

∴ m ∠ROP = 60°

∴ m ∠ROS = 120°

Now, area of the intersecting region = 2(area of sector O-RPS – area of ∆ORS)
                                                         = 2(area of sector O-RPS – area of ∆PRO) [area of ∆PRO = area of ∆ORS]

Area of sector O - RPS = 120360(π) = π3

Area of âˆ†PRO = 34(12)     

∴ Area of the intersecting region = 2(π3) - 2(34) = 2π3-32

Hence, option (e).

CAT 2008 QA Q19: Two circles, both of radii 1 cm, intersect such that the circumference of each one passes through the centre o — Solution | TheCATExam