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Locus

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The Ellipse

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General Form

Intersections of Lines & Conics

Summary&Test

 

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UNIT 9 : THE CONICS

LESSON 3: THE ELLIPSE

 

Definition: Given two fixed points in the plane F1 and F2. An ellipse is the locus (set) of points P such that the sum |PF1+ PF2 | is a constant. The two fixed points are the foci.

The centre is the midpoint of the line segment joining the two foci (F1F2).

 

Text Box: Main Ideas:
	The sum |PF1+ PF2 | = constant = 2a 
	A2A1 is the Major Axis and  |A2A1| = 2a
	B2B1 is the Minor Axis and  |B2B1| = 2b
	
	The standard form of the equation of an ellipse, centre (0, 0), foci on the x-axis:is:
		                               
	The standard form of the equation of an ellipse, centre (0, 0), foci on the y-axis:is:
		                               
	The standard form of the equation of an ellipse, centre (h, k), major axis parallel to the x-axis:is:
		                               
	The standard form of the equation of an ellipse, centre (h,k), major axis parallel to the y-axis:is:
		                               
	                                
	Note a > b always for the ellipse.
	For the ellipse the Pythagorean relation  a2 = b2 + c2  holds true.
 


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 1:

 

Solution:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 3:

 

 

 

 

Solution:

 

 

 

 

Example 4:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 5:

The arch of an overpass is in the shape of a semi-ellipse. If the span of the ellipse (distance between end points of the arch) is 40 m & the height is 8 m,

find the height of the arch at a point 4 m from the end of the arch.

 

Solution:

 

 

 

 

 

 

 

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