Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 1 - Graphs - 1.4 Circles - 1.4 Assess Your Understanding - Page 38: 33

Answer

Centre (3, -2); radius = 5; x-intercepts are 3 + √21 and 3 - √21 y-intercepts are 2 and -6

Work Step by Step

2x² + 2y² - 12x + 8y - 24 = 0 x² + y² - 6x + 4y - 12 = 0 (x² - 6x)+ (y² + 4y) - 12 = 0 (x² - 6x)+ (y² + 4y) - 12 + 12 = 0 + 12 Add 12 to both sides. (x² - 6x + 9)+ (y² + 4y + 4) = 12 + 13 Complete the square of each expression in parenthesis. (x - 3)² + (y + 2)² = (5)² Compare this equation with the equation (x - h)² + (y - k)² = r². The comparison yields the information about the circle. We see that h = 3, k = - 2 and r = 5. The circle has center (3, - 2) and a radius of 5 units. To find the x-intercepts, if any, let y = 0 and solve for x. (x - 3)² + (0 + 2)² = (5)² (x - 3)² + 4 = 25 (x -3)² + 4 - 4 = 25 - 4 Subtract 4 from both sides (x - 3)² = 21 x - 3 = ± √21 x - 3 + 3 = ± √21 + 3 x = 3 ± √21 The x-intercepts are 3 + √21 and 3 - √21. To find y-intercepts, if any, let x = 0 and solve for y. (0 - 3)² + (y + 2)² = 25 9 + (y + 2)² = 25 (y + 2)² + 9 - 9 = 25 - 9 Subtract 9 from both sides. (y + 2)² = 16 y + 2 = ± 4 If y + 2 = 4 y + 2 - 2= 4 - 2 y = 2 If y + 2 = - 4 y + 2 - 2 = - 4 - 2 Subtract 2 from both sides. y = - 6 The y- intercepts are 2 and - 6.
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