JEE Mathematics/ Lakshmikanta Satapathy/ 2D Geometry part 13/ Equation of Ellipse in cartesian and parametric forms derived with the related concepts with explanation of the parameter

1. Physics Helpline
L K Satapathy
Ellipse / Parametric Form
2D Geometry 13
O B D
C
A
P


2. Physics Helpline
L K Satapathy
2D Geometry 13
1 2 2PF PF a 
It is locus of a point in a plane such that the sum of its distances from two
fixed points (foci) in the plane is constant (= length of major axis)
Ellipse :
O
P(x ,y)
F1F2
A1A2
B2
B1
2 2 2 2
( ) ( ) 2x c y x c y a      
   
2 2
2 2 2 2
( ) 2 ( )x c y a x c y      
2 2 2 2 2 2 2
( ) 4 ( ) 4 ( )x c y a x c y a x c y         
2 2 2 2 2
( ) ( ) 4 4 ( )x c x c a a x c y       
1 2 1 2 1 2( ,0) , ( ,0) , ( ,0) , ( ,0), (0, ) , (0, )F c F c A a A a B b B b        
From definition , we have
1 2 1 2 1 22 , 2 , 2F F c A A a B B b  

3. Physics Helpline
L K Satapathy
2D Geometry 13
 
2
2 2 2 2
( )cx a a x c y      
2 2 2 4 2 2 2 2 2 2 2
2 2c x a cx a a x a c a cx a y      
2 2 4 2 2 2 2 2 2
c x a a x a c a y    
2 2 2 2 2 2 2 2
( ) ( )a c x a y a a c    
22
2 2 2
1
yx
a a c
  

22
2 2
1
yx
a b
   [Major axis along x-axis]
22
2 2
1
yx
b a
   [Major axis along y-axis]
2 2 2
4 4 4 ( )cx a a x c y    
O F1F2
A1A2
B2
B1
1 1 1 2
1 1 1 2
2B F B F a
B F B F a
 
  
1 2OF OF c 
2 2 2
1 1 1 1
2 2 2
OB B F OF
b a c
 
  
When P is at B1

4. Physics Helpline
L K Satapathy
2D Geometry 13
Parametric Equation of Ellipse :
Consider two concentric circles of radii a and b
centered at the origin as shown in the figure
Let the coordinates of P = (x ,y)
 OD = x & AB = PD = y
 OC = a and OA = b
Draw ordinates AB and CD at A & C respectively
Drop perpendicular AP upon CD.
We will show that the locus of point P is an ellipse
Construction :
O B D
C
A
P


5. Physics Helpline
L K Satapathy
2D Geometry 13
Let  COD = 
cos & sin
yOD x AB
OC a OA b
     
cos & sinx a y b   
OAB and OCD are similar OA AB
OC CD
 
2 2 2
2
22 2
( )y b a xb y
a aa x

   

2 22 2
2 2 2 2
1 1 . . . (1)
y yx x
b a a b
     
 Point P (x , y) lies on an ellipse
O B D
C
A
P

N.B. : It may be noted that () is NOT the angle subtended by P(x , y) at O
with major axis = a & minor axis = b
[which satisfies equation (1)]

6. Physics Helpline
L K Satapathy
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