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CAT 2003 Slot 1 — QA Question 30

2 CirclesEasy
Passage / Data

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

There are two concentric circles such that the area of the outer circle is four times the area of the inner circle. Let A, B and C be three distinct points on the perimeter of the outer circle such that AB and AC are tangents to the inner circle. If the area of the outer circle is 12 square centimetres then the area (in square centimetres) of the triangle ABC would be

Answer & solution

  • A

    π12

  • B

    9π

  • 93π

  • D

    63π

Solution

Consider the diagram below, as per the given conditions.

Let r and R be the radii of the inner circle and outer circle respectively.

As area of the outer circle is 4 times the area of the inner circle, we have, R = 2r

In âˆ†OAM, sin θ = OMOA=r2r=12

∴ ∠OAM = θ = 30°

Similarly,

∠OAM = ∠OBM = ∠OAC = ∠OCA = 30°

∠OBC = ∠OCB = 30°

∴ ∠BAC = ∠ACB = ∠CBA

∴ ∆ABC is an equilateral triangle.

AB = 2 × (OA2-OM2)=23r

Area of âˆ†ABC = 34 × (AB)2 = 34 × 4 × 3 × r2 = r3r2 ...(i)

But, area of the outer circle = π(2r)2 = 4πr2 = 12

∴ r2 = 3π

∴ Area of âˆ†ABC = 33×3π=93π

Hence, option (c).

CAT 2003 Slot 1 QA Q30: There are two concentric circles such that the area of the outer circle is four times the area of the inner ci — Solution | TheCATExam