Math Problem Statement

Find the equation of the circle shown. Write the equation in center-radius (or standard) form.

Solution

To find the equation of the circle, we need the following pieces of information from the graph:

  1. Center of the circle (h, k): The center can be identified as the point around which the circle is symmetric.
  2. Radius (r): This is the distance from the center to any point on the circle.

From the image:

  • The center of the circle appears to be at (6,0)(6, 0).
  • The radius can be observed as the distance from the center (6,0)(6, 0) to the point where the circle crosses the x-axis at x=9x = 9, so the radius r=3r = 3.

The general equation of a circle in center-radius form is:

(xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2

Substituting the values h=6h = 6, k=0k = 0, and r=3r = 3:

(x6)2+y2=9(x - 6)^2 + y^2 = 9

So, the equation of the circle is:

(x6)2+y2=9(x - 6)^2 + y^2 = 9

Would you like more details or have any other questions?

Here are 5 related questions you can explore:

  1. How do we derive the center-radius form of a circle's equation?
  2. What is the relationship between the general form and the center-radius form of a circle's equation?
  3. How can we find the radius from the diameter or the circumference?
  4. Can this equation be converted into the general form of a circle equation?
  5. How can you determine the equation of a circle given three points on the circle?

Tip: Always double-check the coordinates of the center and the radius by counting grid squares carefully to ensure accuracy in reading the graph!

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Math Problem Analysis

Mathematical Concepts

Geometry
Circles
Coordinate Geometry

Formulas

Equation of a circle: (x - h)^2 + (y - k)^2 = r^2

Theorems

Circle Equation Theorem

Suitable Grade Level

Grades 8-10