Question

If $S$ is the circumcentre of $△ABC$, where $A=(1,3),B=(-3,5),C=(5,-1)$, then

• A. coordinates of $S$ will be $(-8,-10)$
• B. coordinates of $S$ will be $(-4,-5)$
• C. radius of circumcircle will be $5\sqrt{10}$ units
• D. radius of circumcircle will be $5\sqrt{5}$ units

Answer 1

Answer: (A), (C)

The correct options are
A coordinates of $S$ will be $(-8,-10)$
C radius of circumcircle will be $5\sqrt{10}$ units

Ler $S(x,y)$ be the circumcentre of $△ABC$.
We know $SA=SB=SC$
$\Rightarrow S{A}^2=S{B}^2=S{C}^2$
$\Rightarrow (x-1{)}^2+(y-3{)}^2=(x+3{)}^2+(y-5{)}^2=(x-5{)}^2+(y+1{)}^2$
$\Rightarrow -2x-6y+10=6x-10y+34=-10x+2y+26$
$\Rightarrow 8x-4y+24=\mathrm{0.........}\left(1\right)$
$16x-12y+8=\mathrm{0............}\left(2\right)$
solving $\left(1\right)$ and $\left(2\right)$, we get $x=-8,y=-10$
$\therefore S\equiv (-8,-10)$
Now radius of circumcircle will be equal to $SA=SB=SC=R$
$SA=\sqrt{81+169}=\sqrt{250}=5\sqrt{10}$ units