Math Problem Statement

What is the equation of the circle given r = sqrt(2) and center (h, k) = (0, 3)?

Solution

The image shows the equation for a circle, with the following key pieces of information:

  • r=2r = \sqrt{2}, representing the radius of the circle.
  • (h,k)=(0,3)(h, k) = (0, 3), representing the center of the circle.

In general, the equation of a circle in standard form is:

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

Substituting the values given:

  • h=0h = 0
  • k=3k = 3
  • r=2r = \sqrt{2}

The equation of the circle becomes:

(x0)2+(y3)2=(2)2(x - 0)^2 + (y - 3)^2 = (\sqrt{2})^2

Simplifying:

x2+(y3)2=2x^2 + (y - 3)^2 = 2

This is the equation of the circle with radius 2\sqrt{2} and center at (0,3)(0, 3).

Would you like more details or have any questions?

Here are 5 related questions:

  1. How do you derive the equation of a circle from the general formula?
  2. What is the significance of the center (h,k)(h, k) in the equation of a circle?
  3. How would you graph the circle x2+(y3)2=2x^2 + (y - 3)^2 = 2?
  4. What happens if the radius is increased or decreased in the equation of a circle?
  5. Can you describe how to find the area of the circle given the radius r=2r = \sqrt{2}?

Tip: Always remember to square the radius when using it in the equation of a circle.

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Math Problem Analysis

Mathematical Concepts

Geometry
Equations of Circles

Formulas

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

Theorems

Equation of a Circle

Suitable Grade Level

Grades 9-11