Question

Find the centre and radius of each of the following circles:

x² + y² − x + 2y − 3 = 0.

Answer 1

Let (h, k) be the centre of a circle with radius a.
Thus, its equation will be

\[\left( x - h \right)^2 + \left( y - k \right)^2 = a^2\]

Given:

\[x^2 + y^2 - x + 2y - 3 = 0\]

The given equation can be rewritten as follows:

\[\left( x - \frac{1}{2} \right)^2 + \left( y + 1 \right)^2 - \frac{1}{4} - 1 - 3 = 0\]

\[\Rightarrow \left( x - \frac{1}{2} \right)^2 + \left( y + 1 \right)^2 = \frac{17}{4}\]

Thus, the centre is  \[\left( \frac{1}{2}, - 1 \right)\]   and and the radius is \[\frac{\sqrt{17}}{2}\].