Analytical Geometry Parabola Definition A parabola is the locus ( - - PowerPoint PPT Presentation

Analytical Geometry

Parabola

A parabola is the locus (راسم) of a point P(x,y) moving in a plane such that: The distance from P(x,y) to the focus F = The distance from P(x,y) to the directrix D.

Definition

F D

Standard Forms a Parabola

(i) X- Parabola:

2

4 y ax 

' D D

vertix

axis a a focus

directrix

V

2a 2a

F

x y

(0,0) ( ,0) : : V F a directrix x a axis y

•  



Latus Rectum length= 4 a

Standard Forms a Parabola

(i) Y- Parabola:

2

4 x a y 

(0,0) (0, ) : : V F a directrix y a axis x

•  



Latus Rectum length= 4 a

D

' D

vertix

axis a a focus

directrix

V

2a

F

x y 2a

All Standard Forms a Parabola

2

4 y ax 

2

4 y ax  

2

4 x ay 

2

4 x ay  

Example:

Find the vertix, axis, focus, directrix, ends of L.R. and sketch the Parabola

2

8 x y  

Solution:

2

8 x y  

2 a  

Vertix Focus Ends of L.R Axis Directrix

V(0,0) F(0,-2) (4,-2) (-4,-2) x=0 y=2

Parabola with Vertex at V(x0 ,y0)

   

2

4 y y a x x    

   

2

4 x x a y y    

X- Parabola Y- Parabola

General Equation

2

a x cx dy e    

2

by cx dy e    

X- Parabola Y- Parabola

Example:

Find the vertix, axis, focus, directrix, ends of L.R. and sketch the Parabola

Solution:

2

2 16 4 30 y x y    

2

8 2 15 y x y    

2

( 2 ) 8 15 y y x    

2

( 1) 1 8 15 y x     

2

( 1) 8 16 y x    

 

2

( 1) 8 2 y x    

 

2

( 1) 8 2 y x    

a = 2

Vertex Focus Ends of L.R Axis Directrix

(2,1) (0,1) (0,5) (0,-3) y=1 x=4

(0,1) F

(2,1) V

directix axis y x X 4 x  1 y 

* The distance between the focus and the directrix =2a.

* The distance between the vertex and the directrix = the distance between the vertex and the focus = a. * The axis of symmetry and the directrix are perpendicular. * The axis of symmetry passes through the vertex and the focus. * To get the equation of parabola , we must know

• - The type
• - The vertex
• - The value of a

Notes:

Find the equation of the parabola that has vertex V(-4,1) and has focus (-1,1).

Example: Solution

• - The type
• - The vertex
• - The value of a

V(-4,1)

V F

a=3 X-parabola (+)

   

2

1 12 4 y x   

Find the equation of the parabola that has focus (-1,1) and its equation of directrix is y=3.

Example: Solution

• - The type
• - The vertex
• - The value of a

V(-1,2)

F

a=1 y-parabola (-)

   

2

1 4 2 x y    

D