# Find the Coordinates of the Centre of the Circle Passing Through the Points P(6, –6), Q(3, –7) and R (3, 3). - Geometry

#### Question

Find the coordinates of the centre of the circle passing through the points P(6, –6), Q(3, –7) and R (3, 3).

#### Solution

Let the centre be P(a, b)
RO = PO = QO

${RO}^2 = {PO}^2$

$\Rightarrow \left( 3 - a \right)^2 + \left( 3 - b \right)^2 = \left( 6 - a \right)^2 + \left( - 6 - b \right)^2$

$\Rightarrow 9 + a^2 - 6a + 9 + b^2 - 6b = 36 + a^2 - 12a + 36 + b^2 + 12b$

$\Rightarrow 6a - 18b = 54$

$\Rightarrow a - 3b = 9 . . . . . \left( 1 \right)$

${PO}^2 = {OQ}^2$

$\left( 6 - a \right)^2 + \left( - 6 - b \right)^2 = \left( 3 - a \right)^2 + \left( - 7 - b \right)^2$

$\Rightarrow 36 + a^2 - 12a + 36 + b^2 + 12b = 9 + a^2 - 6a + 49 + b^2 + 14b$

$\Rightarrow 3a + b = 7 . . . . . \left( 2 \right)$

Multiplying (2) with 3 we get

$a - 3b = 9 . . . . . \left( 3 \right)$

$9a + 3b = 21 . . . . . \left( 4 \right)$

$\text { Solving }\left( 3 \right) \text { and } \left( 4 \right)$

$a = 3, b = - 2$

Is there an error in this question or solution?