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  • 1. 1.4 Ellipse
    • Another conic section formed by a plane intersecting a cone
    • Ellipse formed when
  • 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. Graph of an Ellipse Note various parts of an ellipse
  • 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. 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. 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. 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. 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. 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.
    • Example 1
    • Find the equation for the ellipse that has its centre at the origin with vertices V (0,± 7) and Foci ( 0,± 2 ).
    • Solution
    • The standard equation of an ellipse is
    where ;
  • 11.
    • Since the vertices are ( 0,± 7 ), we conclude that a = 7. Since the Foci are (0,±2), we have c = 2 .
    • = 22 + 72
    • = 4 + 49
    • = 53
    and equation of the ellipse is
  • 12. Example 2
    • Find the equation for the ellipse that has its centre at the
    • origin with vertices V (0,± 5) and minor axis of length
    • 3. Sketch the ellipse.
    • Solution
    • The standard equation of an ellipse is
    where ; Since the vertices are ( 0,± 5 ), we conclude that b = 5. Since the minor axis is of length 3, we have
  • 13. And equation of the ellipse is (0, 5) (0, – 5 ) 0 y x
  • 14. Example 3
    • Find the focus and equation of the ellipse
    • with centre (0,0) vertices at (2,0) and
    • (0,4).
    • Solution
    From the above and
  • 15.
    • Equation of ellipse is
    • and Foci is ( 0, ) and
  • 16. Example 4
    • Find the centre an vertices of the minor axis and the Foci of the ellipse .
    • Solution
    • The equation of an ellipse is
    For equation , , The centre of the ellipse is ; b = 2 , a = 3 .
  • 17.
    • Vertices of the minor axis are and
    • Foci of the ellipse are and
    • Since , c 2 = a 2 - b 2
    • = 9 – 4
    • = 5
  • 18. Example 5
    • Write the equation of the ellipse that has vertices at and and Foci at and
    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.
    • h = and k =
    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.
    The equation of the ellipse is
  • 21. Example 6
    • 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.
    • Sketch the ellipse.
  • 22. Solution
    • The equation for an ellipse with centre ( h,k ) and the major axis running parallel with the y axis is
    • where ( b ² > a ² )
    • The length of the major axis is 10 units and the minor axis is 6 units.
    • We get 2 b = 10 , 2 a = 6
    • b = 5 , a = 3
  • 23.
    • The equation of the ellipse is
    (3,6) (-3,-4) A . . y x F 1 ( 3,5) B F 2 ( -3,-3) D C ( 3,1) E .
  • 24. Example 7
    • Find the equation of ellipse with vertices ( 8,5 ) and ( 10,1 ) with centre ( 8,k ).
    • Solution
    • Sketching the vertices of the ellipse given.
  • 25. (10,1) (8,5) ( x,y ) ( x 1 , y 1 ) x y
  • 26.
    • We get the centre of ellipse is ( 8,1) , k = 1
    •  x = 8, x 1 = 6, y 1 = 1
    • So equation of ellipse is
    +
  • 27. Example 8
    • Sketch the graph of the equation,
    • Solution:
    • Complete the squares for the expressions
    • 16( x 2 + 4 x +4 ) + 9( y 2 – 2 y + 1 ) = 71 + (16)(4) + (9)(1)
    • 16 ( x + 2 ) 2 + 9 ( y – 1 ) 2 = 144
  • 28.
    • The equation is an ellipse with centre
    • c ( -2,1) and a = 3, b = 4
    • c 2 = b 2 – a 2
    • = 16 – 9
    • = 7
    • c = ±
    • Foci are
  • 29. (-2,5) (1,1) (-5,1) (-2,1) x y Graph for equation