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

Analytical Geometry

e  1 Ellipse Definition An ellipse is the locus ( راسم ) of a point P(x,y) moving in a plane such that: Distance from P x y ( , ) to afo cu s   1 e Distan c e f r om P x y to it ( , ) s di rectrix F D

Standard Forms of Ellipse (i) X- Ellipse: equations:   2 2 x y   2 2 2 1   b a e 1 2 2 a b y b ae ae V  F  F V x C a a a e / a e /    x a e / / x a e

Standard Forms of Ellipse (ii) Y- Ellipse: equations: y 2 2 x y  y a e /   1 2 2 b a v F a e / a ae C x ae a a e / F  b V    y a e /

Notes * The length of the major axis as 2a and the length of b the minor axis as 2b. b a a * 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 2 b * The length of lutus rectum is a

Example: Find the center, vertices, axes, foci, directrices, and sketch the ellipse   2 2 49 x 25 y 1225 Solution: 2 2 x y       7, 5, 0.7 1 a b e 25 49 Center Vertices Foci Axes Directrix (0,0) (0,7) (0,4.9) x=0 y=10 (0,-7) (0,-4.9) y=0 y=-10

Ellipse with Center at C(x 0 ,y 0 ) X- Ellipse Y- Ellipse          2  2  2  2 x x y y x x y y     0 0 0 0 1 1 2 2 2 2 a b b a General Equation      2 2 ax by 2 g x 2 f y c 0  a b

Example: Find the center, vertex, axis, focus, directrix for the ellipse      2 2 5 9 10 54 41 0 x y x y Solution:          2 2 5 x 10 x 9 y 54 y 41 0          2 2 5 x 2 x 9 y 6 y 41 0      2    2    5 x 1 5 9 y 3 81 41 0          2  2  2   2  x 1 y 3 5 x 1 9 y 3 45   1 9 5

      2 2 x 1 y 3   1 9 5 2    a 3, b 5, e 3 Center Vertex Focus Axis Directrix (1,3) (4,3) (3,3) y=3 x=5.5 (-2,3) (-1,3) x=1 x= -3.5

Example: Find the center, vertex, axis, focus, directrix for the ellipse      2 2 9 4 36 8 4 0 x y x y Solution:          2 2 9 x 36 x 4 y 8 y 4 0          2 2 9 x 4 x 4 y 2 y 4 0      2    2    9 x 2 36 4 y 1 4 4 0          2  2  2   2  x 2 y 1 9 x 2 4 y 1 36   1 4 9

     2  2 x 2 y 1   1 4 9 5    a 3, b 2, e 3 Center Vertex Focus Axis Directrix (-2,1) (-2,4) (-2,3.24) y=1 y=5.02 (-2,-2) (-2,-1.24) x=-2 y= -3.02

* To get the equation ellipse , we must know Note: -- The type -- The center -- The value of a, b Example: 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. Solution:      2  2 x 2 y 1 X-ellipse -- The type   1 25 16 -- The center (2,-1) -- The value of a, b a=5, b=4

Example: Find the equation of ellipse with, vertices (6, 8),(6, -2) and one foci is (6, 5). V Solution F C -- The type y-ellipse -- The center (6,3) V -- The value of a, b 2a=10, a=5      2  2 x 6 y 3   ae=2, e=0.4 1 21 25 b 2 =21

Example: Find the equation of ellipse with, one axis = 18 and the ends points of the other axis are (2, 5), (2, -3). Solution 18 V C -- The type x-ellipse -- The center (2,1) -- The value of a, b 2a=18, a=9      2  2 x 2 y 1   2b=8, b=4 1 81 16