Math Problem Statement

The equation of a circle in the xy-plane is given by x^2 - 4x + y^2 + 2y = 31. What are the coordinates of the center of the circle?

Solution

To find the center of the circle given by the equation:

x24x+y2+2y=31x^2 - 4x + y^2 + 2y = 31

we can rewrite this equation in standard form by completing the square for both xx and yy.

Step 1: Completing the Square

  1. For the xx-terms:

    • The terms involving xx are x24xx^2 - 4x.
    • To complete the square, take half of 4-4, square it, and add it inside the square. (4/2)2=4(-4/2)^2 = 4.
    • Rewrite: x24x=(x2)24x^2 - 4x = (x - 2)^2 - 4.
  2. For the yy-terms:

    • The terms involving yy are y2+2yy^2 + 2y.
    • To complete the square, take half of 22, square it, and add it inside the square. (2/2)2=1(2/2)^2 = 1.
    • Rewrite: y2+2y=(y+1)21y^2 + 2y = (y + 1)^2 - 1.

Step 2: Substitute Back

The equation becomes:

(x2)24+(y+1)21=31(x - 2)^2 - 4 + (y + 1)^2 - 1 = 31

Simplify by moving constants to the other side:

(x2)2+(y+1)2=31+4+1(x - 2)^2 + (y + 1)^2 = 31 + 4 + 1

(x2)2+(y+1)2=36(x - 2)^2 + (y + 1)^2 = 36

Step 3: Identify the Center and Radius

The equation is now in the form (xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2, where (h,k)(h, k) is the center of the circle and rr is the radius.

Here:

  • The center (h,k)(h, k) is (2,1)(2, -1).
  • The radius rr is 36=6\sqrt{36} = 6.

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:

  1. How do you complete the square for a general quadratic equation?
  2. What is the geometric interpretation of the circle's center and radius?
  3. How can you derive the radius from a circle's equation in standard form?
  4. What other transformations can be applied to circle equations?
  5. 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