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. [4 MARKS]

Answer 1

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

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

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

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

$\Rightarrow 4x-2y+5=0\dots \dots \left(1\right)$

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

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

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

$\Rightarrow 6x-4y+13=0\dots \dots \left(2\right)$

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 is

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

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