SECTION 10.3 The Ellipse 701 ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 10.3 Assess Your Understanding 1. The distance d from P 2, 5 1 ( ) = − to P 4, 2 2 ( ) = − is d = . (p. 14) 2. To complete the square of x x3 , 2 − add . (p. A29) 3. Find the intercepts of the equation y x 16 4 . 2 2 = − (pp. 20–21) 4. The point symmetric with respect to the y -axis to the point 2, 5 ( ) − is . (pp. 21–23) 5. To graph y x 1 4, 2 ( ) = + − shift the graph of y x2 = to the ( ) left right unit(s) and then ( ) up down unit(s). (pp. 112–120) 6. The standard equation of a circle with center at 2, 3 ( ) − and radius 1 is . (pp. 48–52) Concepts and Vocabulary 7. A(n) is the collection of all points in a plane the sum of whose distances from two fixed points is a constant. 8. Multiple Choice For an ellipse, the foci lie on a line called the . (a) minor axis (b) major axis (c) directrix (d) latus rectum 9. Interactive Figure Exercise Exploring the Graph of an Ellipse Open the “Ellipse” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Uncheck the box “Equation of Ellipse”. Grab the point F ,1 which represents a focus of an ellipse, and move the point F1 to 2, 2 . ( ) − Grab the point F ,2 which represents the other focus of an ellipse, and move the point F2 to 6, 4 . ( ) What is the center of the ellipse? Write your answer as an ordered pair. Confirm the center of the ellipse is at the midpoint of the two foci. (b) Click “Show Trace”. Grab Point A and move it around the Cartesian Plane. Complete a trace of the entire ellipse. What is the sum of the distances from the foci to any point on the ellipse? (c) Hit the reset button ( ) on the interactive figure (upper-right corner). Uncheck the box “Equation of Ellipse”. Move the point F1 to 4, 0 ( ) − and move the point F2 to 4, 0 . ( ) Click “Show Ellipse”. The value of a is defined as the distance from the center of the ellipse to either vertex ( ) V or V . 1 2 What is the value of a? (d) From the question in part (c), notice that the sum of the distances from the foci to any point on the ellipse is a2 . What is the value of c, the distance from the center to either focus? (e) The major axis is the line drawn through the foci of the ellipse. We define b as the distance from the center to points on the ellipse directly above and below the center (when the major axis is parallel to, or on, the x -axis). What is the value of b? (f) How are a, b, and c related to each other? • a b c 2 2 2 + = • a c b 2 2 2 + = • a c b 2 2 2 − = (g) Click “Equation of Ellipse”. Notice the value of a2 is the denominator of the x -term 2 and the value of b2 is the denominator of the y -term. 2 Now move F 1 to 0, 3 ( ) and F2 to 0, 3. ( ) − What is the value of a? What is the value of c? What is the value of b? (h) What is the equation of the ellipse drawn in part (g)? What do you notice about the denominators of the x -term 2 and y -term? 2 10. Interactive Figure Exercise Exploring the Graph of an Ellipse Open the “Ellipse” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) Check the box “Show Ellipse”. Move F1 to 1, 3 ( ) − − and F2 to 7, 3. ( ) − Look at the equation of the ellipse and note the center.What is the value of a? What is the value of b? What is the major axis? Note the denominators in the equation of the ellipse. (b) Move F1 to 2, 3 ( ) and F2 to 2, 3. ( ) − Look at the equation of the ellipse and note the center. What is the value of a? What is the value of b? What is the major axis? Note the denominators in the equation of the ellipse. (c) If the major axis is parallel to the x -axis, then the equation of the ellipse is of the form: • x h a y k b 1 2 2 2 2 ( ) ( ) − + − = • x h b y k a 1 2 2 2 2 ( ) ( ) − + − = 11. For the ellipse x y 4 25 1, 2 2 + = the vertices are the points and . 12. For the ellipse x y 25 9 1, 2 2 + = the value of a is , the value of b is , and the major axis is the -axis. 13. If the center of an ellipse is 2, 3 , ( ) − the major axis is parallel to the x -axis, and the distance from the center of the ellipse to a vertex is a 4 = units, then the coordinates of the vertices are and . 14. Multiple Choice If the foci of an ellipse are 4, 4 ( ) − and 6, 4 , ( ) then the coordinates of the center of the ellipse are . (a) 1, 4 ( ) (b) 4, 1 ( ) (c) 1, 0 ( ) (d) 5, 4 ( ) 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
RkJQdWJsaXNoZXIy NjM5ODQ=