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10-2 Circles Warm Up Find the slope of the line that connects each pair of points. 1 1. (5, 7) and (–1, 6) 6 2. (3, –4) and (–4, 3) –1 Holt Algebra 2 10-2 Circles Warm Up Find the distance between each pair of points. 3. (–2, 12) and (6, –3) 17 4. (1, 5) and (4, 1) 5 Holt Algebra 2 10-2 Circles Objectives Write an equation for a circle. Graph a circle, and identify its center and radius. Holt Algebra 2 10-2 Circles Vocabulary circle tangent 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. Holt Algebra 2 10-2 Circles Example 1: Using the Distance Formula to Write the Equation of a Circle Write the equation of a circle with center (–3, 4) and radius r = 6. Use the Distance Formula with (x2, y2) = (x, y), (x1, y1) = (–3, 4), and distance equal to the radius, 6. Use the Distance Formula. Substitute. Square both sides. Holt Algebra 2 10-2 Circles Check It Out! Example 1 Write the equation of a circle with center (4, 2) and radius r = 7. Use the Distance Formula with (x2, y2) = (x, y), (x1, y1) = (4, 2), and distance equal to the radius, 7. Use the Distance Formula. Substitute. Square both sides. Holt Algebra 2 10-2 Circles Notice that r2 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. Holt Algebra 2 10-2 Circles Helpful Hint If the center of the circle is at the origin, the equation simplifies to x2 + y2 = r2. Holt Algebra 2 10-2 Circles Example 2A: Writing the Equation of a Circle Write the equation of the circle. the circle with center (0, 6) and radius r = 1 (x – h)2 + (y – k)2 = r2 Equation of a circle (x – 0)2 + (y – 6)2 = 12 Substitute. x2 + (y – 6)2 = 1 Holt Algebra 2 10-2 Circles Example 2B: Writing the Equation of a Circle Write the equation of the circle. the circle with center (–4, 11) and containing the point (5, –1) Use the Distance Formula to find the radius. (x + 4)2 + (y – 11)2 = 152 Substitute the values into the equation of a circle. (x + 4)2 + (y – 11)2 = 225 Holt Algebra 2 10-2 Circles Check It Out! Example 2 Find the equation of the circle with center (–3, 5) and containing the point (9, 10). Use the Distance Formula to find the radius. (x + 3)2 + (y – 5)2 = 132 Substitute the values into the equation of a circle. 2 2 (x + 3) + (y – 5) = 169 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 < r2. The points outside the circle satisfy the inequality (x – h)2 + (y – k)2 > r2. Holt Algebra 2 10-2 Circles Example 3: Consumer Application 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)? 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 < 42. Points B, C, D and E are within a 4-mile radius . Check Point F(–2, –3) is near the boundary. 2 2 2 (–2 + 1) + (–3 – 1) < 4 (–1)2 + (–4)2 < 42 1 + 16 < 16 x Point F (–2, –3) is not inside the circle. Holt Algebra 2 10-2 Circles Check It Out! Example 3 What if…? Which homes are within a 3-mile radius of a restaurant located at (2, –1)? The circle has a center (2, –1) and radius 3. The points inside the circle 2 will satisfy the inequality (x – 2) + (y + 1)2 < 32. Points C and E are within a 3-mile radius . Check Point B (1, 2) is near the boundary. 2 2 2 (1 – 2) + (2 + 1) < 3 (–1)2 + (3)2 < 32 1+9<9 x Point B (1, 2) is not inside the circle. 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. Remember! To review linear functions, see Lesson 2-4. Holt Algebra 2 10-2 Circles Example 4: Writing the Equation of a Tangent Write the equation of the line tangent to the circle x2 + y2 = 29 at the point (2, 5). Step 1 Identify the center and radius of the circle. From the equation x2 + y2 = 29, the circle has center of (0, 0) and radius r = . 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. Use the slope formula. Substitute (2, 5) for (x2 , y2 ) and (0, 0) for (x1 , y1 ). 5 The slope of the radius is 2 . Because the slopes of perpendicular lines are 2 negative reciprocals, the slope of the tangent is – 5 . Holt Algebra 2 10-2 Circles Example 4 Continued Step 3 Find the slope-intercept equation of the tangent by using the point (2, 5) and the slope m=– 2.5 Use the point-slope formula. 2 Substitute (2, 5) (x1 , y1 ) and – 5 for m. Rewrite in slope-intercept form. Holt Algebra 2 10-2 Circles Example 4 Continued The equation of the line that is tangent to x2 + y2 = 29 at (2, 5) is . Check Graph the circle and the line. Holt Algebra 2 10-2 Circles Check It Out! Example 4 Write the equation of the line that is tangent to the circle 25 = (x – 1)2 + (y + 2)2, at the point (1, –2). Step 1 Identify the center and radius of the circle. From the equation 25 = (x – 1)2 +(y + 2)2, the circle has center of (1, –2) and radius r = 5. Holt Algebra 2 10-2 Circles Check It Out! Example 4 Continued Step 2 Find the slope of the radius at the point of tangency and the slope of the tangent. Use the slope formula. Substitute (5, –5) for (x2 , y2 ) and (1, –2) for (x1 , y1 ). –3 The slope of the radius is 4 . Because the slopes of perpendicular lines are negative reciprocals, the slope of the tangent is . Holt Algebra 2 10-2 Circles Check It Out! Example 4 Continued Step 3. Find the slope-intercept equation of the tangent by using the point (5, –5) and the slope . Use the point-slope formula. 4 Substitute (5, –5 ) for (x1 , y1 ) and 3 for m. Rewrite in slope-intercept form. Holt Algebra 2 10-2 Circles Check It Out! Example 4 Continued The equation of the line that is tangent to 25 = (x – 1)2 + (y + 2)2 at (5, –5) is . Check Graph the circle and the line. Holt Algebra 2

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