Analytical Geometry e 1 Ellipse Definition An ellipse is the - - PowerPoint PPT Presentation

Analytical Geometry

Ellipse

An ellipse is the locus (راسم) of a point P(x,y) moving in a plane such that:

Definition

F D

1 e 

1 ( , ( , ) ) Distance from e Distan P x y to it c s e f di P x y to afo rectrix

• m

s r cu  

Standard Forms of Ellipse equations:

(i) X- Ellipse:

2 2 2 2

1 x y a b  

C V

V 

F

F  / x a e   / x a e  / a e / a e

ae ae a a b

x

y

 

2 2 2

1 b a e  

Standard Forms of Ellipse equations:

(ii) Y- Ellipse:

2 2 2 2

1 x y b a  

/ a e / a e

C

v

V 

F

F 

ae ae a a b

x

y

/ y a e 

/ y a e  

Notes

b b a a * The length of the major axis as 2a and the length of the minor axis as 2b. * The center of the ellipse is the midpoint of the major axis. * The vertices are the end points of the major axis. * The foci of the ellipse are on the major axis.

2

2b a

* The length of lutus rectum is

Example:

Find the center, vertices, axes, foci, directrices, and sketch the ellipse

2 2

49 25 1225 x y  

Solution:

2 2

1 25 49 x y  

7, 5, 0.7 a b e    

Center Vertices Foci Axes Directrix

(0,0) (0,7) (0,-7) (0,4.9) (0,-4.9) x=0 y=0 y=10 y=-10

Ellipse with Center at C(x0 ,y0)

   

2 2 2 2

1 x x y y a b    

   

2 2 2 2

1 x x y y b a    

X- Ellipse Y- Ellipse

General Equation

2 2

2 2 ax by g x f y c     

a b 

Example:

Find the center, vertex, axis, focus, directrix for the ellipse

Solution:

2 2

5 9 10 54 41 x y x y     

   

2 2

5 10 9 54 41 x x y y     

   

2 2

5 2 9 6 41 x x y y     

   

2 2

5 1 5 9 3 81 41 x y       

   

2 2

5 1 9 3 45 x y    

   

2 2

1 3 1 9 5 x y    

   

2 2

1 3 1 9 5 x y    

Center Vertex Focus Axis Directrix

(1,3) (4,3) (-2,3) (3,3) (-1,3) y=3 x=1 x=5.5 x= -3.5

2 3, 5, 3 a b e   

Example:

Find the center, vertex, axis, focus, directrix for the ellipse

Solution:

2 2

9 4 36 8 4 x y x y     

   

2 2

9 36 4 8 4 x x y y     

   

2 2

9 4 4 2 4 x x y y     

   

2 2

9 2 36 4 1 4 4 x y       

   

2 2

9 2 4 1 36 x y    

   

2 2

2 1 1 4 9 x y    

   

2 2

2 1 1 4 9 x y    

Center Vertex Focus Axis Directrix (-2,1) (-2,4) (-2,-2) (-2,3.24) (-2,-1.24) y=1 x=-2 y=5.02 y= -3.02

5 3, 2, 3 a b e   

* To get the equation ellipse , we must know

• - The type
• - The center
• - The value of a, b

Note:

Write the equation of the ellipse with center at (2,-1), with major axis =10 and parallel to the x- axis, and with minor axis=8.

Example: Solution:

• - The type
• - The center
• - The value of a, b

(2,-1) X-ellipse a=5, b=4

   

2 2

2 1 1 25 16 x y    

Find the equation of ellipse with, vertices (6, 8),(6, -2) and one foci is (6, 5).

Example: Solution

• - The type
• - The center
• - The value of a, b

(6,3)

V F

2a=10, a=5 y-ellipse

   

2 2

6 3 1 21 25 x y    

V C

ae=2, e=0.4 b2=21

Find the equation of ellipse with, one axis = 18 and the ends points of the other axis are (2, 5), (2, -3).

Example: Solution

• - The type
• - The center
• - The value of a, b

(2,1) 2a=18, a=9 x-ellipse

   

2 2

2 1 1 81 16 x y    

V C

2b=8, b=4

18