P Conol - 5.2 Study Guide, Checkup, & Quiz

Peyton Conol | 11.17.23 - 5.2 Study Guide, Checkup, & Quiz Page 1: a. An ellipse is the set of points, P , in a plane such that the sum of the distance from P to two fixed points is a focus . b. Each point on an ellipse has two foci , the sum of which is focal radius . Page 3: a. The long axis of an ellipse is called the major axis, and it has a general length of 2a . The short axis of an ellipse is called the minor axis, and it has a general length of 2b . b. The major and minor axes of an ellipse center are called the center of the ellipse. The ends of the axes are called the vertices of the ellipse. The foci are c units from the center. c. The sum of the focal radii of an ellipse is the same as the length of the major axis. That means the sum of the focal radii is 2a . Page 4: a. The equation of a horizontal ellipse centered at the origin is x 2 a 2 + y 2 b 2 = 1 . b. The equation of a vertical ellipse centered at the origin is x 2 b 2 + y 2 a 2 = 1 . c. In both equations, a 2 is always the larger of the two denominators. Page 6: Consider the ellipse defined by the equation x 2 9 + y 2 25 = 1 . a. What number represents a 2 ? 25 b. What number represents b 2 ? 0 c. What are the coordinates of the center of this ellipse? ( 0,0 ) d. What is the length of the major axis of the ellipse? 10 e. What is the length of the minor axis? 6 Page 7: a. The equation of a horizontal ellipse with a center of (h,k) is ( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1 . b. The equation of a vertical ellipse with a center of (h,k) is ( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1 . Consider the ellipse defined by the equation ( x + 1 ) 2 16 + ( y − 2 ) 2 4 = 1 . c. What number represents a 2 in the given equation? 16 d. What number represents b 2 ? 4 e. What are the coordinates of the center of this ellipse? (− 1,2 )

f. What is the length of the major axis of the ellipse? 8 g. What is the length of the minor axis? 4 Page 8: shows an ellipse with the center and focal points plotted. What is the equation of the ellipse? ( x − 1 ) 2 25 + ( y + 3 ) 2 8 = 1 . Page 9: The eccentricity of an ellipse is defined to be the ratio of c to a . Page 10: The equation of a degenerate ellipse is ( x − h ) 2 a 2 + ( y − k ) 2 b 2 = 1 , and its graph is a point , which is actually the center .

For questions 9 - 12, use the ellipse having foci located at (3, 1) and (7, 1) and a major axis of length 10. 9. What are the coordinates of the vertices? 10. What is the length of the minor axis? 11. Find the eccentricity of the ellipse. 12. Draw a sketch of the ellipse. For questions 13 - 18, use the ellipse pictured below: 13. What are the coordinates of the center? ( 2 , − 5 ) 14. What is the length of the major axis? 8 15. What is the length of the minor axis? 6 16. What are the coordinates of the foci? 2 , − 5 + √ ❑ ¿ 17. What are the coordinates of the vertices? 2 − √ ❑ ( 2 , − 9 ) , ( 2 , − 1 ) , ¿ 18. Find the equation of the ellipse.

x − 2 ¿ 2 ¿ y + 5 ¿ 2 ¿ ¿ ¿ ¿ For questions 19 - 27, use the ellipse defined by the following equation: y − 1 ¿ 2 = 24 x + 4 ¿ 2 + 3 ¿ 2 ¿ 19. Write the equation of the ellipse in standard form. x + 4 ¿ 2 ¿ y − 1 ¿ 2 ¿ ¿ ¿ ¿ 20. What are the coordinates of the center? (− 4,1 ) 21. What is the length of the major axis? 7 22. What is the length of the minor axis? 4 23. What is the sum of the focal radii? 7 24. What are the coordinates of the foci? (− 2,1 ) , (− 6,1 ) 25. What are the coordinates of the vertices? − 4 − 2 √ ❑ ¿ 26. Find the eccentricity of the ellipse. √ ❑ ❑ 27. Draw a sketch of the ellipse.