cat 2020 Question 19

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

Accepted Answer

Equation of circle $x^2 + y^2 + 2gx + 2fy + c = 0 $

It passes through $ (0,0),(4,0),(3,9). $

$ \Rightarrow ext{Substitute } (0,0):\ c = 0 $

$ \Rightarrow ext{Substitute } (4,0):\ 16 + 0 + 8g + 0 = 0 \ ;\ g = -2 $

$ \Rightarrow ext{Substitute } (3,9):\ 9 + 81 - 12 + 18f = 0 \ ;\ f = -13/3 $

$ ext{Radius of circle } r = \sqrt{g^2 + f^2 - c} $

$ \Rightarrow r^2 = \frac{205}{9} $

Therefore, Area = $\pi r^2 = \frac{205\pi}{9} $

Question 19.