Question

Find the circumcenter of the triangle whose sides are
$3x-y-5=0,x+2y-4=0,5x+3y+1=0$

Answer 1

A

$3x-y-5=0$

$\Rightarrow y=3x-5$

B

$x+2y-4=0$

$y=\frac{-1}{2}x+2$

C

$5x+3y+1=0$

$y=\frac{-5}{3}x-\frac{1}{3}$

D

$3x-5=\frac{-1}{2}x+2$

$\Rightarrow D(2,1)$

E

$3x-5=\frac{-5}{3}x-\frac{1}{3}$

$\Rightarrow E(1,-2)$

F

$\frac{-1}{2}x+2=\frac{-5}{3}x-\frac{1}{3}$

$\Rightarrow F(-2,3)$

Use these point to make a circle.

$D=(2-h{)}^2+(1-k{)}^2=r^2$

$E=(1-h{)}^2+(-2-k{)}^2=r^2$

$F=(-2-h{)}^2+(3-k{)}^2=r^2$

$\Rightarrow h=\frac{-6}{7},k=\frac{2}{7},r=\frac{5\sqrt{17}}{7}$

Circumcenter$=(\frac{-6}{7},\frac{2}{7})$