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# Math1.3

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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