Question

Show that the equation x² + y² – 4x + 6y – 5 = 0 represents a circle. Find its centre and radius.

Answer 1

The general equation of a conic is as follows 

ax² + 2hxy + by² + 2gx + 2fy + c = 0 where a, b, c, f, g, h are constants 

For a circle, a = b and h = 0. 

The equation becomes: 

x² + y² + 2gx + 2fy + c = 0…(i) 

Given, x² + y² – 4x + 6y – 5 = 0

Comparing with (i) we see that the equation represents a circle with 2g = - 4

⇒ g = - 2,

2f = 6 

⇒ f = 3 and c = - 5.

Centre ( - g, - f) = { - ( - 2), - 3} = (2, - 3). 

Radius = \(\sqrt{g^2+f^2-c}\) 

=   \(\sqrt{(-2)^2+3^2-(-5)}\) 

= \(\sqrt{4+9+5}\) = \(\sqrt{18}\) = \(3\sqrt{2}\)