Question

Find the equation of the circle circumscribing the triangle formed by the points $(0,0),(3,4)$ and $(-4,3)$

Answer 1

Let equation be $x^2+y^2+2gx+2fy+c=0$

Passes through $(0,0)\Rightarrow c=0$

Passes through $(3,4)$

$\Rightarrow 9+16+6g+8f=0$

$\Rightarrow 6g+8f=-25$

Passes through $(-4,3)$

$\Rightarrow -8g+6f=-25$

$\Rightarrow 14g-2f\Rightarrow f=-7g$

$\Rightarrow 6g-56g=-25$

$\Rightarrow g=1/2\Rightarrow f=-7/2$

$\therefore$ eqn of circle is $x^2+y^2+x-7y=0$