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Geometry 10.6 Equations of Circles Geometry Geometry Objectives • Write the equation of a circle. • Use the equation of a circle and its graph to solve problems. Geometry Finding Equations of Circles • You can write an equation of a circle in a coordinate plane if you know its radius and the coordinates of its center. 6 y (x, y) 4 (h, k) 2 x 5 -2 Geometry Finding Equations of Circles • Suppose the radius is r and the center is (h, k). • Let (x, y) be any point on the circle. • The distance between (x, y) and (h, k) is r, so • We use Distance Formula. • (Told you it wasn’t going away). 6 y (x, y) 4 (h, k) 2 x 5 -2 Finding Equations of Circles ( x h) ( y k ) r Geometry 2 2 • Square both sides to find the standard equation of a circle with radius r and center (h, k). (x – h)2 + (y – k)2 = r2 If the center is at the origin, then the standard equation is x 2 + y 2 = r 2. 6 y (x, y) 4 (h, k) 2 x 5 -2 Geometry Writing a Standard Equation of a Circle • Write the standard equation of the circle with a center at (-4, 0) and radius 7 (x – h)2 + (y – k)2 = r2 Standard equation of a circle. [(x – (-4)]2 + (y – 0)2 = 72 Substitute values. (x + 4)2 + (y – 0)2 = 49 Simplify. Geometry Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle. r= ( x2 x1 ) 2 ( y2 y1 ) 2 Use the Distance Formula r= (5 1) 2 (1 2) 2 Substitute values. r= (4) 2 (3) 2 Simplify. r= r= 16 9 25 Simplify. Addition Property Square root the result. r=5 Geometry Ex. 2: Writing a Standard Equation of a Circle The point (1, 2) is on a circle whose center is (5, 1). Write a standard equation of the circle. (x – h)2 + (y – k)2 = r2 Standard equation of a circle. [(x – 5)]2 + [y –(-1)]2 = 52 Substitute values. (x - 5)2 + (y + 1)2 = 25 Simplify. Geometry Graphing Circles • If you know the equation of a circle, you can graph the circle by identifying its center and radius. Geometry Graphing a circle • The equation of a • (x+2)2 + (y-3)2 = 9 circle is • [x – (-2)]2 + (y – 3)2=32 (x+2)2 + (y-3)2 = 9. • The center is (-2, 3) and Graph the circle. the radius is 3. First rewrite the equation to find the center and its radius. Geometry Graphing a circle • To graph the circle, place the point of a compass at (-2, 3), set the radius at 3 units, and swing the compass to draw a full circle. 6 4 2 -5 -2 Geometry Graphs of Circles 1. Rewrite the equation to find the center and radius. The center is at (13, 4) and the radius is 4. – (x – h)2 + (y – k)2= r2 – (x – 13)2 + (y – 4)2= 42 – (x - 13)2 + (y - 4)2 = 16 Geometry •The center is (1, -3) •Radius = 3 • Geometry •The center is (-4, -3) •Radius = 1 • Geometry •(x – h)2 + (y – k)2 = r2 •(x – 8)2 + (y + 6)2 = 42 •(x – 8)2 + (y +6)2 =16 • Find Radius 3.14 = 3.14 * r2 • r=1 •(x – h)2 + (y – k)2 = r2 •(x + 12)2 + (y +11)2 = 12 •(x +12)2 + (y +11)2 =1 Geometry • Find Radius • Find Radius 12 * 3.14 = 2 *3.14 * r 8 * 3.14 = 2 *3.14 * r • r = 16 • r=4 •(x – h)2 + (y – k)2 = r2 •(x – h)2 + (y – k)2 = r2 •(x + 9)2 + (y +9)2 = 62 •(x + 13)2 + (y -4)2 = 42 •(x +12)2 + (y +11)2 =36 •(x +13)2 + (y - 4)2 =16 Geometry •The center is (4, 0) •Radius = 3 •(x – h)2 + (y – k)2 = r2 •(x - 4)2 + (y + 0)2 = 42 •(x – 4)2 + y 2 =16 Geometry •The center is (-3,-4) •Radius = 2 •(x – h)2 + (y – k)2 = r2 •(x +3)2 + (y + 4)2 = 22 •(x + 3)2 + (y+4) 2 =4 Geometry •The center is (2,4) •Radius = √6 = 2.5 • Geometry •The center is (4,3) •Radius = √2 = 1.4 •