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The Hyperbola

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Summary&Test

 

 

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 UNIT 9 : THE CONICS

 LESSON 5: THE HYPERBOLA

 

 

Definition: Given two fixed points in the plane F1 and F2. A hyperbola is the locus (set) of points P such that the difference |PF2 PF1 | 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: See diagrams at left and above.
	The sum |PF2 -  PF1 | = constant = 2a 
	A2A1 is the Transverse Axis and  |A2A1| = 2a
	B2B1 is the Conjugate Axis and  |B2B1| = 2b
	
	The standard form of the equation of a hyperbola, centre (0, 0), foci on the x-axis:is:
		                             
	The standard form of the equation of a hyperbola, centre (0, 0), foci on the y-axis:is:
	
	                             
	The standard form of the equation of a hyperbola, centre (h, k), major axis parallel to the x-axis:is:
	
                                    
	The standard form of the equation of a hyperbola, centre (h,k), major axis parallel to the y-axis:is:
	
                                 
	                              
	For the hyperbola the Pythagorean relation  c2 = a2 + b2  holds true.
 
       

	 
 

 

 

 

 Hyperbola with foci on the x-axis:

 

 

Hyperbola with foci on the y-axis:

 

 

 

 

 

 

 

 

 

Example 1:

 

 

Solution:

 

 

 

 

 

 

 

 

 

 

 

Example 2:

Given the hyperbola with equation 9x2 - 4y2 = - 36.

a) Put the equation in standard form and find the

values of a, b, c, e.

b) Determine the coordinates of the vertices.

c) State the length of the transverse and conjugate axes.

d) Find the equations of the asymptotes.

e) Draw the graph.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Example 3:

 

 

  

Solution:

 

 

 

 

 

 

 Example 4:

Given the hyperbola with equation 9x2 54x 25y2 + 200y 544 = 0.

a) Put the equation in standard form and find the

values of a, b, c, e.

b) Determine the coordinates of the vertices.

c) State the length of the transverse and conjugate axes.

d) Find the equations of the asymptotes.

e) Draw the graph

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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