Geometry Equations of a Circle

Goals Write the equation of a circle. Use the equation of a circle to graph the circle on the coordinate plane. Solve problems with circles. 4/20/2017

Circle Definition A circle is the set of points on a plane that are equidistant from the center. The radius, r, is the distance between the center (h, k) and any point (x, y) on the circle. (x, y) r (h, k) 4/20/2017

Circle Equation Use the Distance Formula to write this. (x, y) r (h, k) 4/20/2017

Circle Equation (x, y) Square both sides: r (h, k) 4/20/2017

The Equation of a Circle Where: (h, k) is the center r is the radius (x, y) is any point on the circle (x, y) r (h, k) 4/20/2017

What is the center and radius? (x – 9)2 + (y – 1)2 = 25 Center: (9, 1) Radius: 5 (x – 9)2 + (y – 1)2 = 52 4/20/2017

What is the center and radius? (x – 2)2 + (y + 1)2 = 1 (x – 2)2 + (y – (-1))2 = 12 Center: (2, -1) Radius: 1 4/20/2017

What is the center and radius? (x – 6)2 + y2 = 100 Center: (6, 0) Radius: 10 (x – 6)2 + (y – 0)2 = 102 4/20/2017

Your Turn Identify the center and radius of each circle: (x – 12)2 + (y + 3)2 = 4 Center: (12, –3) Radius = 2 x2 + y2 = 121 Center: (0, 0) Radius = 11 4/20/2017

Example 42 = (x – 5)2 + (y – 6)2 16 = (x – 5)2 + (y – 6)2 Write the equation of a circle with center (5, 6) and radius 4. 42 = (x – 5)2 + (y – 6)2 16 = (x – 5)2 + (y – 6)2 or (x – 5)2 + (y – 6)2 = 16 4/20/2017

Your Turn 82 = (x – 1)2 + (y – (-3))2 (x – 1)2 + (y + 3)2 = 64 Write the equation of a circle with center (1, -3) and radius 8. 82 = (x – 1)2 + (y – (-3))2 (x – 1)2 + (y + 3)2 = 64 4/20/2017

What if we don’t know r? The point (3, 2) is on a circle with center (5, 4). Write the equation. 4/20/2017

What if we don’t know r? r2 = (3 – 5)2 + (2 – 4)2 r2 = (–2 )2 + (–2)2 The point (3, 2) is on a circle with center (5, 4). Write the equation. r2 = (3 – 5)2 + (2 – 4)2 r2 = (–2 )2 + (–2)2 r2 = 4 + 4 = 8 DON’T SIMPLIFY! 4/20/2017

Write the equation. r2 = 8 (x – 5)2 + (y – 4)2 = 8 The point (3, 2) is on a circle with center (5, 4). Write the equation. r2 = 8 (x – 5)2 + (y – 4)2 = 8 4/20/2017

Your Turn. r2 = (-1 – 2)2 + (4 – 3)2 r2 = (-3)2 + (1)2 r2 = 9 + 1 = 10 The point (-1, 4) is on a circle with center (2, 3). Write the equation. r2 = (-1 – 2)2 + (4 – 3)2 r2 = (-3)2 + (1)2 r2 = 9 + 1 = 10 (x – 2)2 + (y – 3)2 = 10 4/20/2017

Graphing Circles Graph the circle given by the equation (x – 2)2 + (y – 1)2 = 9 First find the center (h, k). What is h? 2 What is k? 1 4/20/2017

Graphing Circles continued (x – 2)2 + (y – 1)2 = 9 Center (2, 1) What is r? 3 Why? (x – 2)2 + (y – 1)2 = 32 4/20/2017

Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. Draw the center. Draw points at the ends of 4 radii. 4/20/2017

Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. Draw the center. Draw points at the ends of 4 radii. Sketch the circle. 4/20/2017

Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. Draw the center. Draw points at the ends of 4 radii. Sketch the circle. 4/20/2017

Your Turn Graph: (x – 1)2 + (y + 3)2 = 16 Solution: Center: (1, -3) Radius: 4 4/20/2017

Problem (x + 1)2 + (y – 1)2 = 25 Is the point (3, 4) on the circle, in its interior, or in the exterior? Directions: Make a sketch of the circle. Then locate (3, 4) and answer the question. 4/20/2017

Graphical Solution On the circle. Graph: (x + 1)2 + (y – 1)2 = 25 Center: (-1, 1) Radius: 5 Locate (3, 4) On the circle. 4/20/2017

What about (3, 2)? In the interior of the circle. 4/20/2017

What about (-5, -3)? In the exterior of the circle. 4/20/2017

You could do this… Find the distance from the center (-1, 1) to the point (-5, -3): Since the distance to the point is larger than the radius, it must be in the exterior of the circle. 5 4/20/2017

What you can now do: Write the equation of a circle. Graph a circle from its equation. Determine where a point is in the interior, exterior, or on a circle. 4/20/2017

Quick Practice Identify the center and the radius of the circle: (x + 2)2 + y2 = 9 Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle. Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 1 4/20/2017

Quick Practice Identify the center and the radius of the circle: (x + 2)2 + y2 = 9 Center (-2, 0) Radius = 3 4/20/2017

Quick Practice Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle. 4/20/2017

Quick Practice Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 1 4/20/2017

Practice Problems 4/20/2017