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Math1.3

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Math1.3

  1. 1. 1.4 Ellipse <ul><li>Another conic section formed by a plane intersecting a cone </li></ul><ul><li>Ellipse formed when </li></ul>
  2. 2. Definition: An ellipse is defined as the set of points in a plane such that the sum of the distances from P to two fixed points is a constant. The two fixed points are the foci.
  3. 3. Graph of an Ellipse Note various parts of an ellipse
  4. 4. The equation of an ellipse with centre (0,0) and foci x y c F 2 (-c.0) F 1 (c,0) V 2 (-a,o) V 1 (a,0) M 1 (0,b) M 2 (0,-b) G H J K
  5. 5. We summarized the properties of the ellipse with the horizontal major axis as, a > b >0 Vertices : Major axis : horizontal, length 2a Minor axis : vertical, length 2b Foci : where c 2 =a 2 -b 2 Latus rectum : vertical length
  6. 6. The equation of an ellipse with center (0,0) and foci x y c F 1 (0,c) F 2 (0,-c) V 2 (0,-b) V 1 (0,b) M 1 (0,a) M 2 (0,-a)
  7. 7. We summarised the properties of this second form of ellipse as follow:- b > a >0 Vertices : Major axis : vertical, length 2b Minor axis : horizontal, length 2a Foci where c 2 =b 2 -a 2 Latus rectum: vertical length
  8. 8. The equation of an ellipse with centre (h,k) and foci a > b >0 Vertices : Major axis : horizontal, length 2a Minor axis : vertical, length 2b Foci : where c 2 =a 2 -b 2 Latus rectum : vertical length
  9. 9. b > a >0 Vertices : Major axis : vertical, length 2b Minor axis : horizontal, length 2a Foci where c 2 =b 2 -a 2 Latus rectum: vertical length The equation of an ellipse with center (h,k) and foci
  10. 10. <ul><li>Example 1 </li></ul><ul><li>Find the equation for the ellipse that has its centre at the origin with vertices V (0,± 7) and Foci ( 0,± 2 ). </li></ul><ul><li>Solution </li></ul><ul><li>The standard equation of an ellipse is </li></ul>where ;
  11. 11. <ul><li>Since the vertices are ( 0,± 7 ), we conclude that a = 7. Since the Foci are (0,±2), we have c = 2 . </li></ul><ul><li> = 22 + 72 </li></ul><ul><li> = 4 + 49 </li></ul><ul><li> = 53 </li></ul>and equation of the ellipse is
  12. 12. Example 2 <ul><li>Find the equation for the ellipse that has its centre at the </li></ul><ul><li>origin with vertices V (0,± 5) and minor axis of length </li></ul><ul><li>3. Sketch the ellipse. </li></ul><ul><li>Solution </li></ul><ul><li>The standard equation of an ellipse is </li></ul>where ; Since the vertices are ( 0,± 5 ), we conclude that b = 5. Since the minor axis is of length 3, we have
  13. 13. And equation of the ellipse is (0, 5) (0, – 5 ) 0 y x
  14. 14. Example 3 <ul><li>Find the focus and equation of the ellipse </li></ul><ul><li>with centre (0,0) vertices at (2,0) and </li></ul><ul><li>(0,4). </li></ul><ul><li>Solution </li></ul>From the above and
  15. 15. <ul><li>Equation of ellipse is </li></ul><ul><li>and Foci is ( 0, ) and </li></ul>
  16. 16. Example 4 <ul><li>Find the centre an vertices of the minor axis and the Foci of the ellipse . </li></ul><ul><li>Solution </li></ul><ul><li>The equation of an ellipse is </li></ul> For equation , , The centre of the ellipse is ; b = 2 , a = 3 .
  17. 17. <ul><li>Vertices of the minor axis are and </li></ul><ul><li>Foci of the ellipse are and </li></ul><ul><li>Since , c 2 = a 2 - b 2 </li></ul><ul><li> = 9 – 4 </li></ul><ul><li> = 5 </li></ul>
  18. 18. Example 5 <ul><li>Write the equation of the ellipse that has vertices at and and Foci at and </li></ul>Solution The vertices and foci are on the same horizontal line . The equation of the ellipse is , Where a > b The centre of the ellipse is at the midpoint of the major axes
  19. 19. <ul><li>h = and k = </li></ul>The distance between the centre and vertex is 5 units ; thus . The distance between the centre ( 2,-5) and focus ( 5,-5) is 3 units, thus c = 3 , = = 16
  20. 20. <ul><li> </li></ul><ul><li> </li></ul>The equation of the ellipse is
  21. 21. Example 6 <ul><li>Find the equation of an ellipse with centre ( 3,1 ) and the major axis running parallel with the y axis. The length of the major axis is 10 units and the minor axis is 6 units. </li></ul><ul><li>Sketch the ellipse. </li></ul>
  22. 22. Solution <ul><li>The equation for an ellipse with centre ( h,k ) and the major axis running parallel with the y axis is </li></ul><ul><li> where ( b ² > a ² ) </li></ul><ul><li> </li></ul><ul><li>The length of the major axis is 10 units and the minor axis is 6 units. </li></ul><ul><li>We get 2 b = 10 , 2 a = 6 </li></ul><ul><li> b = 5 , a = 3 </li></ul>
  23. 23. <ul><li>The equation of the ellipse is </li></ul>(3,6) (-3,-4) A . . y x F 1 ( 3,5) B F 2 ( -3,-3) D C ( 3,1) E .
  24. 24. Example 7 <ul><li>Find the equation of ellipse with vertices ( 8,5 ) and ( 10,1 ) with centre ( 8,k ). </li></ul><ul><li>Solution </li></ul><ul><li>Sketching the vertices of the ellipse given. </li></ul>
  25. 25. (10,1) (8,5) ( x,y ) ( x 1 , y 1 ) x y
  26. 26. <ul><li>We get the centre of ellipse is ( 8,1) , k = 1 </li></ul><ul><li> x = 8, x 1 = 6, y 1 = 1 </li></ul><ul><li>So equation of ellipse is </li></ul>+
  27. 27. Example 8 <ul><li>Sketch the graph of the equation, </li></ul><ul><li>Solution: </li></ul><ul><li>Complete the squares for the expressions </li></ul><ul><li>16( x 2 + 4 x +4 ) + 9( y 2 – 2 y + 1 ) = 71 + (16)(4) + (9)(1) </li></ul><ul><li>16 ( x + 2 ) 2 + 9 ( y – 1 ) 2 = 144 </li></ul>
  28. 28. <ul><li>The equation is an ellipse with centre </li></ul><ul><li>c ( -2,1) and a = 3, b = 4 </li></ul><ul><li>c 2 = b 2 – a 2 </li></ul><ul><li> = 16 – 9 </li></ul><ul><li> = 7 </li></ul><ul><li>c = ± </li></ul><ul><li>Foci are </li></ul>
  29. 29. (-2,5) (1,1) (-5,1) (-2,1) x y Graph for equation

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