Show that the equation ${x^2} + {y^2} + x - y = 0$ represents a circle. Find its centre and radius.

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Show that the equation ${x^2} + {y^2} + x - y = 0$ represents a circle. Find its centre and radius.

Answer:

The general equation of a conic is,

ax² + 2hxy + by² + 2gx + 2fy + c = 0

a, b, c, f, g, h are constants

For a circle, a = b and h = 0.

The equation is,

x² + y² + 2gx + 2fy + c = 0

x² + y² + x – y = 0

The equation represents a circle with 2g = 1

$\begin{array}{l} g = \frac{1}{2}\\ 2f = - 1\\ f = - \frac{1}{2}\\ c = 0\\ Centre( - g, - f) = \left\{ { - \frac{1}{2}, - \left( { - \frac{1}{2}} \right)} \right\}\\ = > \left( { - \frac{1}{2},\frac{1}{2}} \right)\\ Radius = \sqrt {{g^2} + {f^2} - c} \\ = > \sqrt {{{\frac{1}{2}}^2} + ( - {{\frac{1}{2}}^2}) - 0} \\ = > \sqrt {\frac{1}{4} + \frac{1}{4}} \\ = > \sqrt {\frac{1}{2}} \end{array}$