CAT 2020 Slot 3 QA Question & Solution
GeometryMedium
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
Options
$\frac{14\pi}{3}$
$\frac{123\pi}{7}$
$\frac{12\pi}{5}$
$\frac{205\pi}{9}$
Solution
Equation of circle $x^2+y^2+2gx+2fy+c=0$
It passes through (0,0), (4,0) and (3,9). Substitute each point in the above equation:
=> On substituting the value (0,0) in the above equation, we obtain: $c=0$
=> On substituting the value (4,0) in the above equation, we obtain: $16+0+8g+0 = 0$ ; $g=-2$
=> On substituting the value (3,9) in the above equation, we obtain: $9+81-12+18f = 0$ ; $f= -13/3$
Radius of the circle r = $\sqrt{\ g^2+f^2-c}$ => $r^2=\frac{205}{9}$
Therefore, Area = $\pi\ r^2=\frac{205\pi\ }{9}$