Algebra 2 Attributes of Hyperbolas
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More Hyperbolic
Attributes of Hyperbolas

 

 An ellipse is just a stretched out circle.

 

 

 
 If we put a minus sign between the terms of an ellipse
 we get a stretched out hyperbola.
 

 

 
 The hyperbola still approaches diagonal asymptotes
 that have the slopes:
 

 

 
 So for this problem, the slopes are:
 

 

 
 Since we don't have anything added or subtracted inside the parentheses, 
 the hyperbola is centered at the origin (0,0).
 That means the asymptotes are as well.
 
 That makes the asymptote equations Y = 1/2 X and Y = - 1/2 X.
 So the graph is:
 

 

 
 When we worked with ellipses, 
 we spent a lot of time talking about focus points.
 Hyperbolas have focus points too!
 They are on the centerline of the hyperbola.
 You find them using the denominators of the X and Y terms.
 Remember if a term does not have a denominator showing,
 the denominator is 1.
 
 The equation for the distance to the focus point is:
 

 
 So for our hyperbola:
 

 

 
 The distance from the center to the focus points is:
 

 

 
 The focus points are (4.47,0) and (-4.47,0)
 

 

 
 There is a trick you can do with hyperbola focus points:
 
 1) Choose any point on either hyperbola line.
 2) Measure the distance from the point to the further away focus point
 3) Measure the distance to the closer focus point
 4) Subtract the smaller distance from the larger distance
 5) No matter what two points you chose, you always get the same answer!
 

 

 
 The answer will be different for each hyperbola,
 but any point in a hyperbola will give you the same answer
 as any other point on that same hyperbola.
 

   copyright 2005 Bruce Kirkpatrick

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