Area of circumcircle of a triangle

CAT 2020 Slot 3 · QA · Hard · Coordinate Geometry

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The vertices of a triangle are (0,0)(0,0), (4,0)(4,0) and (3,9)(3,9). The area of the circle passing through these three points is

  1. A.

    \frac{14\pi}{3}

  2. B.

    \frac{123\pi}{7}

  3. C.

    \frac{205\pi}{9}

  4. D.

    \frac{12\pi}{5}

Answer

C

Explanation

Let equation of the circle be x2+y2+2gx+2fy+c=0x^2 + y^2 + 2gx + 2fy + c = 0. Passes through (0,0)  ⟹  c=0(0,0) \implies c = 0. Passes through (4,0)  ⟹  16+8g=0  ⟹  g=−2(4,0) \implies 16 + 8g = 0 \implies g = -2. Passes through (3,9)  ⟹  9+81+2(−2)(3)+2f(9)=0  ⟹  90−12+18f=0  ⟹  18f=−78  ⟹  f=−13/3(3,9) \implies 9 + 81 + 2(-2)(3) + 2f(9) = 0 \implies 90 - 12 + 18f = 0 \implies 18f = -78 \implies f = -13/3.

Radius squared R2=g2+f2−c=(−2)2+(−13/3)2=4+1699=2059R^2 = g^2 + f^2 - c = (-2)^2 + (-13/3)^2 = 4 + \frac{169}{9} = \frac{205}{9}. Area of the circle = πR2=205π9\pi R^2 = \frac{205\pi}{9}.

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