Concept

Example 1 Write an Equation Given the Radius LANDSCAPING The plan for a park puts the center of a circular pond of radius 0.6 mile at 2.5 miles east and 3.8 miles south of the park headquarters. Write an equation to represent the border of the pond, using the headquarters as the origin. Since the headquarters is at (0, 0), the center of the pond is at (2.5, –3.8) with radius 0.6 mile. Answer: The equation is (x – 2.5) 2 + (y + 3.8) 2 = (x – h) 2 + (y – k) 2 = r 2 Equation of a circle (x – 2.5) 2 + (y + 3.8) 2 = (h, k) = (2.5, –3.8), r = 0.6 (x – 2.5) 2 + (y + 3.8) 2 = 0.36Simplify.

Example 2 Write an Equation from a Graph Write an equation for the graph.

Example 2 Write an Equation from a Graph (x – h) 2 + (y – k) 2 =r 2 Standard form (7 – 0) 2 + (3 – 3) 2 =r 2 x = 7, y = 3, h = 0, k = 3 (7) 2 + (0) 2 =r 2 Simplify =r 2 Evaluate the exponents. 49=r 2 Add. Answer: So, the equation of the circle is x 2 + (y – 3) 2 = 49.

Example 3 Write an Equation Given a Diameter Write an equation for a circle if the endpoints of the diameter are at (2, 8) and (2, –2). First, find the center of the circle. Midpoint Formula (x 1, y 1 ) = (2, 8), (x 2, y 2 ) = (2, –2) Add. Simplify.

Example 3 Write an Equation Given a Diameter Now find the radius. Simplify.Subtract. (x 1, y 1 ) = (2, 3), (x 2, y 2 ) = (2, 8) Distance Formula The radius of the circle is 5 units, so r 2 = 25. Substitute h, k, and r 2 into the standard form of the equation of a circle.

Example 3 Write an Equation Given a Diameter Answer: An equation of the circle is (x – 2) 2 + (y – 3) 2 = 25.