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+2x-6y+14=0

Verified step by step guidance

1. Start by rewriting the given equation: \(x^2 + y^2 + 2x - 6y + 14 = 0\).
2. Group the \(x\) terms and \(y\) terms together: \((x^2 + 2x) + (y^2 - 6y) + 14 = 0\).
3. Complete the square for the \(x\) terms: take half of 2, which is 1, and square it to get 1. Add and subtract 1 inside the equation.
4. Complete the square for the \(y\) terms: take half of -6, which is -3, and square it to get 9. Add and subtract 9 inside the equation.
5. Rewrite the equation as \((x^2 + 2x + 1) - 1 + (y^2 - 6y + 9) - 9 + 14 = 0\), then simplify and express it in the form \((x + 1)^2 + (y - 3)^2 = r^2\) 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.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions in a perfect square form. This technique is essential to transform the given equation into the standard circle form by grouping x and y terms separately.

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.