11.2.4 Circles in the Coordinate Plane Key Objectives
• Write equations and graph circles in the coordinate plane. • Use the equation and graph of a circle to solve problems. Theorems, Postulates, Corollaries, and Properties • Equation of a Circle The equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 . Example 1 Writing the Equation of a Circle The equation of a circle with center ( h , k ) and radius r is ( x − h ) 2 + ( y − k ) 2 = r 2 .
The equation for a circle is determined in this example. The coordinates of the center of the circle and the radius are given. Substitute the given values into the equation for a circle. Simplify the equation. The equation for the circle is ( x + 1) 2 + ( y − 5) 2 = 16.
The equation for a circle is determined in this example. The coordinates of the center of the circle and the coordinates of a point the circle passes through are given. Begin by determining the length of the radius for the circle. A radius is a line segment from the center of the circle to any point on the circle. Find the length of the radius using the Distance Formula. Substitute the given values for the center of the circle and the point on the circle into the distance formula and solve for r . The length of the radius is 5 units. To find the equation for the circle, substitute the value for the radius and the coordinates of the center of the circle into the equation. Simplify the equation. The equation for the circle is ( x − 2) 2 + y 2 = 25.
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