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. See Examples 3–5. x^2+y^2-2x+12y-12=0
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 a circle from a general equation, one must rearrange it into this form. This involves completing the square for both x and y terms, which reveals the center and radius directly.
Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is essential for rewriting the equation of a circle in standard form. By adding and subtracting the necessary constants, one can isolate the squared terms and identify the center and radius of the circle.
Solving Quadratic Equations by Completing the Square
Types of Graphs: Circle, Point, or Nonexistent
When analyzing equations, they can represent different types of graphs: a circle, a single point, or no graph at all. A circle exists if the equation can be rearranged into standard form with a positive radius. A point occurs when the radius is zero, indicating a single coordinate. If the equation leads to a contradiction, it is considered nonexistent.