MATH 30 – WCELLIPSE 1. For each of the following, determine the center of the ellipse and the endpoints of each axis. x2 y2 a. 1 64 36 x2 y2 b. 1 16 49 2 2 x 7 y 5 c. 1 4 25 x 3 y 8 2 d. 2 9 100 e. 9x 2 16y 2 144 1 f. 49x 2 7y 1 36 2 2 Tough Question! 2. Sketch the graph of each of the ellipses in question 1 and check your graph on a graphing calculator. (Write the equation you need to put in your calculator) 3. Write the equation of each of the ellipses below. a. b. c. Page 1 f. shorter axis of length 2 8. Place Stadium has an air–filled fabric dome roof that forms the shape of an ellipse when viewed from above. For each ellipse. (10.4. e. d. Find an equation for this semi–ellipse. 5) with the longer axis of length 12 and parallel to the x–axis. a. determine the coordinates of the center. and (3. center at (3. 3x 2 y2 6x 8y 11 0 x 2 121y 2 726y 968 0 9x 2 25y 2 9x 50y 197. center at (2. b. Its maximum length is approximately 230 m. Taking a cross section of the roof at its greatest width results in a semi–ellipse.75 0 16x 2 4y 2 96x 8y 84 0 4x 2 9y2 96x 16x 18y 11 0 36x 2 64y 2 108x 128y 431 0 6. (3. –5). 1). 2). c. x 2 y 1 1 9 25 2 a. 2). 4) with the longer axis of length 8 and parallel to the y–axis. 2). Write the equation of an ellipse with a center (3. 230 m 190 m b. and its maximum height is approximately 60 m. 64 36 2 1 7. x 3 y 1 2 b. shorter axis of length 10 b. Find the equation of an ellipse satisfying the given conditions: a.C. a. Change the following to general form. B. passing through (4. 60 m 190 m Page 2 . the endpoints of the both axes. 5. Find an equation for the ellipse formed by the base of the roof. its maximum width is approximately 190 m. The promoters of a concert plan to send fireworks up from a point on the stage that is 30 m lower than the center in part b. How far is that point on the stage from the roof? 9. Then find the standard form of the equation of each ellipse.c. b) a) d) c) f) e) Page 3 . Describe the transformations (translations and stretches) that have been applied to the unit circle x 2 y 2 1 to produce each of the ellipses below. and 40 m along the major axis of this ellipse from its center. Endpts(0.–7) 1 16 49 2 2 x 7 y 5 c) C(–7.0).5).(–5.(3. Endpts(–9.(0.0).(6.(0.0).(–7.1 . .(0.(0.–2). Endpts(4.(–8. 2 2 x 4 y 2 a) 1 9 1 b) x 3 c) x 4 2 4 25 2 y 2 2 1 9 y 4 4 2 1 Page 4 . Write the equation of each of the ellipses below.0).0).(–4. Endpts(8.(–4.7). determine the centre of the ellipse and the endpoints of each axis.0).8). Endpts(4.5). x2 y2 a) C(0. For each of the following.Ellipse Worksheet Answers 1.1 f) 49x 2 7y 1 36 C(–2.5).–3) 6 6 2 2 20 8 .1 7 7 7 7 2.6).(–7.0).10) 1 4 25 x 3 y 8 2 d) 2 1 9 100 e) 9x 2 16y 2 144 C(3. Endpts .3).–6) 1 64 36 x2 y2 b) C(0.8).0).(3.0).1).8).18) C(0. 2.(0. (Write the equation you need to put in your calculator) 3. 2.0). Sketch the graph of each of the ellipses in question 1 and check your graph on a graphing calculator.1 .(0. the endpoints of the both axes.(0. 1 10.1) 36x 2 64y 2 108x 128y 431 0 C(–1.(0. Change the following to general form. passing through (–4.4) with the longer axis of length 8 and parallel to the y–axis. shorter axis of 2 2 x 3 y 4 length 2 1 1 16 8.75 0 C(0. 4 . shorter axis of 2 2 x 2 y 5 length 10.3).1).5.C. Endpts(–1.1).5. Endpts(–5. 230 m x2 y2 1 13225 9025 190 m b) Taking a cross section of the roof at its greatest width results in a semi–ellipse. Endpts(–4.1).1). a) 3x 2 y2 6x 8y 11 0 C(–1.1). (10.5.–1). 1 49 9 5. 4 .4) 6.5.4.5.–2). Endpts 1 b) c) d) e) f) 2 10.(2. Write the equation of an ellipse with a centre (3.(–3. For each ellipse.5.5) with the longer axis of length 12 and parallel to the x–axis.(11.1).1).4) C(–3. Endpts(–1. 2 y 1 a) x 1 9 25 2 2 x 3 y 1 b) 1 64 36 2 25 x 2 9 y 2 18 y 216 0 36 x 2 64 y 2 216 x 128 y 1916 0 7. 1. Its maximum length is approximately 230 m. and its maximum height is approximately 60 m. Endpts(–11. Place Stadium has an air–filled fabric dome roof that forms the shape of an ellipse when viewed from above.–3).(–3. 4 30 2 C(0.5.(–5. its maximum width is approximately 190 m. B. determine the coordinates of the centre.(–1.1).5.(5.5) 16x 2 4y 2 96x 8y 84 0 2 2 4x 9y 96x 16x 18y 11 0 C(2.5. 1. and 2 2 x 3 y 2 (3.1).1).(2. Find the equation of an ellipse satisfying the given conditions: a) Centre at (2.–2).5.–2).(5.2).5).–1).(0.4). 4 30 . Find an equation for this semi–ellipse.3).–1).–2).(0. 1 36 25 b) Centre at (–3. 60 m x2 y2 1 9025 3600 190 m Page 5 .–3).4) x 121y 726y 968 0 2 2 9x 25y 9x 50y 197. a) Find an equation for the ellipse formed by the base of the roof.(2. (–1.–2). (3.3). x2 y 2 1 4 25 Page 6 . up 6. up 6. horizontal stretch by a factor of 2 about the y–axis. and left 3. and left 3. e) 2 Vertical stretch by a factor of 3 about the x–axis.4 m 9. Describe the transformations (translations and stretches) that have been applied to the unit circle x 2 y 2 1 to produce each of the ellipses below. horizontal stretch by a factor of 2 about the y–axis. horizontal stretch by a factor of 2 about the y–axis. b ) a ) d ) c ) f ) e ) a) Vertical stretch by a factor of 3 about the x–axis. b) x 3 4 2 y 6 9 2 1 Vertical stretch by a factor of 3 about the x–axis. Then find the standard form of the equation of each ellipse. d) x 5 x 6 9 2 y 2 4 2 1 Vertical stretch by a factor of 3 about the x–axis. and left 3. How far is that point on the stage from the roof? 84. and 40 m along the major axis of this ellipse from its centre. horizontal stretch by a factor of 2 about the y–axis. and left 3. horizontal stretch by a factor of 2 about the y–axis. up 6. up 6. c) 4 x 4 y 7 2 1 1 2 1 y2 1 4 Vertical stretch by a factor of 3 about the x–axis. and left 3.c) The promoters of a concert plan to send fireworks up from a point on the stage that is 30 m lower than the centre in part b). up 6.