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

#### Solution

Let P (x, y) be the centre of the circle passing through the points O(0, 0), A(–2,1) and B(–3,2).

Then, OP = AP = BP

Now,

`OP = AP ⇒ OP^2 = AP^2`

`⇒ x^2 + y^2 = (x + 2)^2 + (y – 1)^2`

`⇒ x^2 + y^2 = x^2 + y^2 + 4x – 2y + 5`

⇒ 4x – 2y + 5 = 0 ….(1)

and, OP = BP ⇒ OP^{2} = BP^{2}

`⇒ x^2 + y^2 = (x + 3)^2 + (y – 2)^2`

`⇒ x^2 + y^2 = x^2 + y^2 + 6x – 4y + 13`

⇒ 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, `\text{}Radius=OP=sqrt(x^{2}+y^{2})=\sqrt{\frac{9}{4}+\frac{121}{4}}`

`=\frac{1}{2}\sqrt{130}`