Math Problem Statement
Solution
To find the center of the circle given by the equation:
we can rewrite this equation in standard form by completing the square for both and .
Step 1: Completing the Square
-
For the -terms:
- The terms involving are .
- To complete the square, take half of , square it, and add it inside the square. .
- Rewrite: .
-
For the -terms:
- The terms involving are .
- To complete the square, take half of , square it, and add it inside the square. .
- Rewrite: .
Step 2: Substitute Back
The equation becomes:
Simplify by moving constants to the other side:
Step 3: Identify the Center and Radius
The equation is now in the form , where is the center of the circle and is the radius.
Here:
- The center is .
- The radius is .
Answer
The coordinates of the center of the circle are: D) (2, -1)
Would you like further details on any steps, or do you have questions?
Here are some related questions to consider:
- How do you complete the square for a general quadratic equation?
- What is the geometric interpretation of the circle's center and radius?
- How can you derive the radius from a circle's equation in standard form?
- What other transformations can be applied to circle equations?
- How would you find the diameter of this circle?
Tip: Remember that completing the square is a useful method for converting equations to standard forms, especially for circles and other conic sections.
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Math Problem Analysis
Mathematical Concepts
Geometry
Algebra
Completing the Square
Formulas
(x - h)^2 + (y - k)^2 = r^2 for the equation of a circle in standard form
Theorems
Circle Equation Standard Form
Suitable Grade Level
Grades 9-12
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