In Exercises 109–111, give the center and radius of each circle. x^2 + y^2 - 4x + 2y - 4 = 0

Verified step by step guidance

1. Rewrite the given equation of the circle: $x^2\; +\; y^2\; -\; 4x\; +\; 2y\; -\; 4\; =\; 0$. Group the terms involving $x$ and $y$ together: $(x^2\; -\; 4x)\; +\; (y^2\; +\; 2y)\; =\; 4$.
2. Complete the square for the $x$-terms. Take half the coefficient of $x$ (which is $-4$), square it, and add it inside the parentheses: $(x^2\; -\; 4x\; +\; 4)$. To maintain equality, add $4$ to the right-hand side of the equation.
3. Complete the square for the $y$-terms. Take half the coefficient of $y$ (which is $2$), square it, and add it inside the parentheses: $(y^2\; +\; 2y\; +\; 1)$. To maintain equality, add $1$ to the right-hand side of the equation.
4. Rewrite the equation with the completed squares: $(x\; -\; 2)^2\; +\; (y\; +\; 1)^2\; =\; 9$. This is now in the standard form of a circle: $(x\; -\; h)^2\; +\; (y\; -\; k)^2\; =\; r^2$, where $(h,\; k)$ is the center and $r$ is the radius.
5. Identify the center and radius from the equation. The center is $(h,\; k)\; =\; (2,\; -1)$, and the radius is $r\; =\; \backslash \backslash sqrt\{9\}\; =\; 3$.

Key Concepts

Standard Form of a Circle

The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center 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 variable terms and create a squared term, which simplifies the process of identifying the center and radius of a circle.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0. In the context of circles, the terms involving x and y can be rearranged to form a quadratic equation, which is essential for identifying the geometric properties of the circle, such as its center and radius.