Question

Find the coordinates of the centre of the circle passing through the points (0, 0), (–2, 1) and (–3, 2). Also, find its radius.

Answer 1

Let P (x, y) be the centre of the circle passing through the points 0 (0, 0), A (-2, 1) and B (-3, 2). Then, OP = AP = BP

Now, OP = AP $\Rightarrow$ $O{P}^2=A{P}^2$

$\Rightarrow$ $x^2+y^2=(x+2{)}^2+(y-1{)}^2$

$\Rightarrow$ $x^2+y^2=x^2+y^2+4x-2y+5$

$\Rightarrow$ 4x-2y+ 5 =0 ....(1)

and, OP = BP $\Rightarrow$ $O{P}^2=B{P}^2$

$\Rightarrow$ $x^2+y^2=(x+3{)}^2+(y-2{)}^2$

$\Rightarrow$ $x^2+y^2=x^2+y^2+6x-4y+13$

$\Rightarrow$ 6x - 4y + 13 = 0 .......(2)

On solving equations (1) and (2), we get

x = $\frac{3}{2}$ and y = $\frac{11}{2}$

Thus , the coordinates of the centre are $(\frac{3}{2},\frac{11}{2})$

Now, Radius = OP = $\sqrt{x^2+y^2}$ = $\sqrt{\frac{9}{4}+\frac{121}{4}}$

= $\sqrt{\frac{130}{4}}$ units .