Describe the graph of each equation as a circle, a point, or nonexistent. If it is a circle, give the center and radius. If it is a point, give the coordinates. x²+y²+4x-8y+32=0

Verified step by step guidance

1. Start by rewriting the given equation: \(x^2 + y^2 + 4x - 8y + 32 = 0\).
2. Group the \(x\) terms and \(y\) terms together: \((x^2 + 4x) + (y^2 - 8y) + 32 = 0\).
3. Complete the square for the \(x\) terms: take half of 4, which is 2, and square it to get 4. Add and subtract 4 inside the equation.
4. Complete the square for the \(y\) terms: take half of -8, which is -4, and square it to get 16. Add and subtract 16 inside the equation.
5. Rewrite the equation as \((x^2 + 4x + 4) + (y^2 - 8y + 16) + 32 - 4 - 16 = 0\), then simplify and express it in the form \((x + 2)^2 + (y - 4)^2 = r^2\) to identify the center and radius.

Key Concepts

Standard Form of a Circle Equation

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius. Converting a general quadratic equation into this form helps identify the circle's properties clearly.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in the form (x - h)^2 or (y - k)^2. This technique is essential to transform the given equation into the standard form of a circle.

Determining the Nature of the Graph

After rewriting the equation, the radius squared (r^2) determines the graph's nature: if r^2 > 0, it's a circle; if r^2 = 0, it's a point; if r^2 < 0, the graph does not exist in the real plane.