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

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

• A. Center (0, 0), r = 5
• B. Center (-1, 2), r = 4
• C. Center (1, 1), r = 5
• D. Center (-1, -1), r = 3

Answer: (B)

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

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