# 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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