Find the centre and the radius of the circle represented by the equation $z ¯ z - (2 + 3 i) z - (2 - 3 i) ¯ z + 9 = 0$ .

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Solution

$z ¯ ¯ ¯ z - (2 + 3 i) z - (2 - 3 i) ¯ ¯ ¯ z + 9 = 0$
put $z = x + i y$ $z = x - i y$ $z ¯ ¯ ¯ z = x^{2} + y^{2}$
$x^{2} + y^{2} - (2 + 3 i) (x + i y) - 2 - 3 i) (x - i y) + 9 = 0$
$x^{2} + y^{2} - (2 (x + i y + x - i y)) - 3 i (x + y - x + i y) + 9 = 0$
$x^{2} + y^{2} - 4 x - 3 i (2 i y) + 9 = 0$
$x^{2} + y^{2} - 4 x + 6 y + 9 = 0$
centre $(2, - 3)$ , Radius $= \sqrt{2^{2} + 3^{2} - 9} = 2$

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