1. Holt Algebra 2 10-2 Circles 10-2 Circles Holt Algebra2 Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

2. Holt Algebra 2 10-2 Circles Warm Up Find the slope of the line that connects each pair of points. 1. (5, 7) and (–1, 6) 2. (3, –4) and (–4, 3)

3. Holt Algebra 2 10-2 Circles Warm Up Find the distance between each pair of points. 3. (–2, 12) and (6, –3) 4. (1, 5) and (4, 1)

4. Holt Algebra 2 10-2 Circles Write an equation for a circle. Graph a circle, and identify its center and radius. Objectives

5. Holt Algebra 2 10-2 Circles circle tangent Vocabulary

6. Holt Algebra 2 10-2 Circles A circle is the set of points in a plane that are a fixed distance, called the radius, from a fixed point, called the center. Because all of the points on a circle are the same distance from the center of the circle, you can use the Distance Formula to find the equation of a circle.

7. Holt Algebra 2 10-2 Circles Write the equation of a circle with center (–3, 4) and radius r = 6. Example 1: Using the Distance Formula to Write the Equation of a Circle Use the Distance Formula with (x 2, y 2 ) = (x, y), (x 1, y 1 ) = (–3, 4), and distance equal to the radius, 6. Use the Distance Formula. Substitute. Square both sides.

8. Holt Algebra 2 10-2 Circles Write the equation of a circle with center (4, 2) and radius r = 7. Check It Out! Example 1

9. Holt Algebra 2 10-2 Circles Notice that r 2 and the center are visible in the equation of a circle. This leads to a general formula for a circle with center (h, k) and radius r. If the center of the circle is at the origin, the equation simplifies to x 2 + y 2 = r 2. Helpful Hint

10. Holt Algebra 2 10-2 Circles Write the equation of the circle. Example 2A: Writing the Equation of a Circle (x – 0) 2 + (y – 6) 2 = 1 2 x 2 + (y – 6) 2 = 1 the circle with center (0, 6) and radius r = 1 (x – h) 2 + (y – k) 2 = r 2 Equation of a circle Substitute.

11. Holt Algebra 2 10-2 Circles Use the Distance Formula to find the radius. Substitute the values into the equation of a circle. (x + 4) 2 + (y – 11) 2 = 225 the circle with center (–4, 11) and containing the point (5, –1) (x + 4) 2 + (y – 11) 2 = 15 2 Write the equation of the circle. Example 2B: Writing the Equation of a Circle

12. Holt Algebra 2 10-2 Circles Find the equation of the circle with center (–3, 5) and containing the point (9, 10). Check It Out! Example 2

13. Holt Algebra 2 10-2 Circles The location of points in relation to a circle can be described by inequalities. The points inside the circle satisfy the inequality (x – h) 2 + (y – k) 2 < r 2. The points outside the circle satisfy the inequality (x – h) 2 + (y – k) 2 > r 2.

14. Holt Algebra 2 10-2 Circles Use the map and information given in Example 3 on page 730. Which homes are within 4 miles of a restaurant located at (–1, 1)? Example 3: Consumer Application The circle has a center (–1, 1) and radius 4. The points insides the circle will satisfy the inequality (x + 1) 2 + (y – 1) 2 < 4 2. Points B, C, D and E are within a 4-mile radius. Point F (–2, –3) is not inside the circle. Check Point F(–2, –3) is near the boundary. (–2 + 1) 2 + (–3 – 1) 2 < 4 2 (–1) 2 + (–4) 2 < 4 2 1 + 16 < 16 x

15. Holt Algebra 2 10-2 Circles What if…? Which homes are within a 3-mile radius of a restaurant located at (2, –1)? Check It Out! Example 3

16. Holt Algebra 2 10-2 Circles A tangent is a line in the same plane as the circle that intersects the circle at exactly one point. Recall from geometry that a tangent to a circle is perpendicular to the radius at the point of tangency. To review linear functions, see Lesson 2-4. Remember!

17. Holt Algebra 2 10-2 Circles Write the equation of the line tangent to the circle x 2 + y 2 = 29 at the point (2, 5). Example 4: Writing the Equation of a Tangent Step 1 Identify the center and radius of the circle. From the equation x 2 + y 2 = 29, the circle has center of (0, 0) and radius r =.

18. Holt Algebra 2 10-2 Circles Example 4 Continued Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Substitute (2, 5) for (x 2, y 2 ) and (0, 0) for (x 1, y 1 ). Use the slope formula. The slope of the radius is. 5 2 Because the slopes of perpendicular lines are negative reciprocals, the slope of the tangent is. 2 5 –

19. Holt Algebra 2 10-2 Circles Example 4 Continued Use the point-slope formula. Rewrite in slope-intercept form. Substitute (2, 5) (x 1, y 1 ) and – for m. 2 5 Step 3 Find the slope-intercept equation of the tangent by using the point (2, 5) and the slope m =. 2 5 –

20. Holt Algebra 2 10-2 Circles Example 4 Continued The equation of the line that is tangent to x 2 + y 2 = 29 at (2, 5) is. Check Graph the circle and the line.

21. Holt Algebra 2 10-2 Circles Write the equation of the line that is tangent to the circle 25 = (x – 1) 2 + (y + 2) 2, at the point (5, –5). Check It Out! Example 4 Check Graph the circle and the line.