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10.7 Warm-up. For each equation, make a table and then graph. Identify the y-intercept and the slope. 1. 2. 3. 4. 10.7 Warm-up (Answers). 2. For each equation, make a table and then graph . Identify each y–int . and the slope . 1. 2. 3. 4. 3. 1. 4. Geometry.
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10.7 Warm-up For each equation, make a table and then graph. Identify the y-intercept and the slope. 1. 2. 3. 4. Geometry 10.7 Circles in the Coordinate Plane
10.7 Warm-up (Answers) 2 For each equation, make a table and then graph. Identify each y–int. and the slope. 1. 2. 3. 4. 3 1 4 10.7 Circles in the Coordinate Plane
Geometry 10.7 Circles in the Coordinate Plane
10.7 Essential Question What is the equation of a circle with center (h, k) and radius r in the coordinate plane? Geometry 10.7 Circles in the Coordinate Plane
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. Geometry 10.7 Circles in the Coordinate Plane
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) Geometry 10.7 Circles in the Coordinate Plane
Finding the Equation of a Circle Use the Distance Formula to write this. (x, y) r Square both sides: (h, k) Geometry 10.7 Circles in the Coordinate Plane
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) Geometry 10.7 Circles in the Coordinate Plane
Example 1What is the center and radius? a. (x – 9)2 + (y – 1)2 = 25 Center: (9, 1) Radius: 5 (x – 9)2 + (y – 1)2 = 52 Geometry 10.7 Circles in the Coordinate Plane
Example 1What is the center and radius? b. (x – 2)2 + (y + 1)2 = 1 (x – 2)2 + (y – (-1))2 = 12 Center: (2, -1) Radius: 1 Geometry 10.7 Circles in the Coordinate Plane
Example 1What is the center and radius? c. (x – 6)2 + y2 = 100 Center: (6, 0) Radius: 10 (x – 6)2 + (y – 0)2 = 102 Geometry 10.7 Circles in the Coordinate Plane
Your Turn 1 Identify the center and radius of each circle: a. (x – 12)2 + (y + 3)2 = 4 Center: (12, –3) Radius = 2 b. x2+ y2 = 121 Center: (0, 0) Radius = 11 Geometry 10.7 Circles in the Coordinate Plane
Example 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 Geometry 10.7 Circles in the Coordinate Plane
Your Turn 2 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 Geometry 10.7 Circles in the Coordinate Plane
What if we don’t know r? The point (3, 2) is on a circle with center (5, 4). Write the equation. Geometry 10.7 Circles in the Coordinate Plane
What if we don’t know r? 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! Geometry 10.7 Circles in the Coordinate Plane
Write the equation. The point (3, 2) is on a circle with center (5, 4). Write the equation. r2 = 8 (x – 5)2 + (y – 4)2 = 8 Geometry 10.7 Circles in the Coordinate Plane
Your Turn 3 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 Geometry 10.7 Circles in the Coordinate Plane
Graphing Circles Graph the circle given by the equation (x – 2)2 + (y – 1)2 = 9 Steps: 1. Find the center (h, k). What is h? 2 • What is k? • 1 Geometry 10.7 Circles in the Coordinate Plane
Graphing Circles continued (x – 2)2 + (y – 1)2 = 9 Center (2, 1) 2. Find the radius, r. 3 Why? (x – 2)2 + (y – 1)2 = 32 Geometry 10.7 Circles in the Coordinate Plane
Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 3. Draw the center. 4. Draw points at the ends of 4 radii. Geometry 10.7 Circles in the Coordinate Plane
Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 3. Draw the center. 4. Draw points at the ends of 4 radii. 5. Sketch the circle. Geometry 10.7 Circles in the Coordinate Plane
Graphing Circles Knowing the center is (2, 1) and the radius is 3. Graph the circle. 3. Draw the center. 4. Draw points at the ends of 4 radii. 5. Sketch the circle. Geometry 10.7 Circles in the Coordinate Plane
Your Turn 4 Graph: (x – 1)2 + (y + 3)2 = 16 Solution: Center: (1, -3) Radius: 4 Geometry 10.7 Circles in the Coordinate Plane
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. Geometry 10.7 Circles in the Coordinate Plane
Graphical Solution Graph: (x + 1)2 + (y – 1)2 = 25 Solution: Center: (-1, 1) Radius: 5 Locate (3, 4) On the circle. Geometry 10.7 Circles in the Coordinate Plane
What about (3, 2)? In the interior of the circle. Geometry 10.7 Circles in the Coordinate Plane
What about (-5, -3)? In the exterior of the circle. Geometry 10.7 Circles in the Coordinate Plane
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 Geometry 10.7 Circles in the Coordinate Plane
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. Geometry 10.7 Circles in the Coordinate Plane
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 Geometry 10.7 Circles in the Coordinate Plane
Quick Practice • Identify the center and the radius of the circle: (x + 2)2 + y2 = 9 Center (-2, 0) Radius = 3 Geometry 10.7 Circles in the Coordinate Plane
Quick Practice • Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle. Geometry 10.7 Circles in the Coordinate Plane
Quick Practice • Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 4 Geometry 10.7 Circles in the Coordinate Plane
Homework Geometry 10.7 Circles in the Coordinate Plane