CAT 2020 Quant Question [Slot 3] with Solution 15

Question

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

  1. \(\frac{{14\pi }}{3}\)
  2. \(\frac{{12\pi }}{5}\)
  3. \(\frac{{123\pi }}{7}\)
  4. \(\frac{{205\pi }}{9}\)
Option: 4
Solution:

\(a = 4,b = \sqrt {90} ;c = \sqrt {82} \)

Area of the triangle \( = \frac{1}{2} \times 4 \times 9 = 18\)

The circumradius of the triangle \((R) = \frac{{abc}}{{4\Delta }}\)

Area of the circle \( = \pi {R^2} = \pi {\left( {\frac{{abc}}{{4\Delta }}} \right)^2} = \frac{{\pi {{(4 \cdot \sqrt {90}  \cdot \sqrt {82} )}^2}}}{{{{(4 \cdot 18)}^2}}}\)

\( = \pi  \times \frac{{16 \times 90 \times 82}}{{16 \times 18 \times 18}} = \frac{{205}}{9}\pi \)

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CAT 2020 Quant questions with Solutions