10.7 Write and Graph Equations of Circles Hubarth Geometry

In the circle below, let point (x, y) represent any point on the circle whose center is at the origin. Let r represent the radius of the circle. In the right triangle, r = length of hypotenuse x = length of a leg y = length of a leg By the Pythagorean Theorem, you can write x 2 + y 2 = r 2 This is an equation of a circle with center at the origin. r x y

Ex 1 Write an Equation of a Circle Write an equation of the circle. Solution The radius is 4 and the center is at the origin.

Standard Equation of a Circle If the center of a circle is not at the origin, you can use the Distance Formula to write an equation of the circle. For example, the circle shown at the right has center (3, 5) and a radius of 4. Let (x, y) represent any point on the circle. Use the Distance Formula to find the lengths of the legs. leg: I x-3 I leg: I y-5 I hypotenuse: 4 Use these expressions in the Pythagorean Theorem to find an equation of the circle. (x-3) 2 + (y-5) 2 =4 2 This is an example of the standard equation of a circle.. (3, 5) (x, y) 4 I y-5 I I x-3 I

Standard Equation of a Circle In the coordinate plane, the standard equation of a circle with center at (h, k) and radius r is (x-h) 2 + (y-k) 2 = r 2 x-coordinate of the center y-coordinate of the center. r (x, y) (h, k)

Ex 2 Write the Standard Equation of a Circle. Write the standard equation of the circle with center (2, -1) and radius 3.. (2, -1) Solution

Ex 3 Graph a Circle Graph the given equation of the circle. Solution a. The center is (1, 2) and the radius is 2. b. The center is (-2, 0) and the radius is 2. (1, 2) (-2, 0)

Practice Write an equation of the circle