Give the center and radius of the circle represented by each equation. See Examples 3 and 4. x^2+y^2-4x+12y=-4

Verified step by step guidance

1. Rewrite the given equation in the standard form of a circle, which is \((x-h)^2 + (y-k)^2 = r^2\).
2. Start by rearranging the terms: \(x^2 - 4x + y^2 + 12y = -4\).
3. Complete the square for the \(x\) terms: \(x^2 - 4x\). To do this, take half of the coefficient of \(x\), square it, and add it inside the equation. \((-4/2)^2 = 4\).
4. Complete the square for the \(y\) terms: \(y^2 + 12y\). Similarly, take half of the coefficient of \(y\), square it, and add it inside the equation. \((12/2)^2 = 36\).
5. Add the constants from completing the square to both sides of the equation to maintain equality: \((x^2 - 4x + 4) + (y^2 + 12y + 36) = -4 + 4 + 36\). Now, rewrite the equation in the form \((x-h)^2 + (y-k)^2 = r^2\) to identify the center \((h, k)\) and the radius \(r\).

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is given by (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. To identify the center and radius from a general equation, it is often necessary to rearrange the equation into this standard form through completing the square.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique involves manipulating the equation to isolate the x and y terms, allowing for easier identification of the center and radius of the circle. It is essential for converting the general form of the circle's equation into standard form.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0, where a, b, and c are constants. In the context of circles, the x² and y² terms represent the squared distances from the center, and understanding their properties is crucial for analyzing the geometric representation of the circle in the coordinate plane.