What is the center and radius of the circle defined by the equation x^2 + y^2 + 2x = 4y + 11?

To find the center and radius of the circle defined by the equation ( x^2 + y^2 + 2x = 4y + 11 ), we start by rearranging this equation into the standard form of a circle's equation, which is ( (x - h)^2 + (y - k)^2 = r^2 ). Here, ((h, k)) represents the center and (r) the radius.

First, we rearrange the given equation:

1. Move all terms to one side:

[

x^2 + y^2 + 2x - 4y - 11 = 0

]

2. Rearranging gives us:

[

x^2 + 2x + y^2 - 4y = 11

]

Next, we complete the square for the (x) and (y) terms.

For the (x) terms (x^2 + 2x):

• Take half of the coefficient of (x), which is (1), square it to get (1), and add (1):

[

x^2 + 2x