Algebra 2 Graphing Ellipse Foci
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Don't Lose Your Focus
Graphing Ellipse Foci

 

 There are a couple of interesting details about ellipses.

 
 The orbits of all of the planets moons, space shuttles, asteroids, comets 
 and anything else that's out there orbiting around
 are ellipses
 
 There are two special points inside an ellipse.
 These are called focus points (weird name, I know)
 
 Focus points always live along the stretched out direction centerline
 The focus points are the same distance from the center as each other.
 Like this:

 

 
 There is a trick that will always work on an ellipse.
 Choose any point on the ellipse line.
 Measure the distance from each of the focus points.
 Then add the two measurements together.
 No matter what two points on the ellipse line you choose,
 you will always get the same answer!
 

 

 
 As an ellipse gets stretched out more and more,
 the focus points move further from the center of the ellipse
 and closer to the edge.
 

 

 
 Now these focus points would just be some sort of math trivia
 except that the orbits of planets and moons and satellites and stuff like that
 are all ellipses.
 

 

 
 Now if the orbiting thing in the picture is the earth,
 and you want to draw the sun in the picture too
 where would you put it?
 
 Right in the middle?
 
 NO!
 
 It would be at one of the focus points!
 

 

 
 (Actually, the sun is not centered EXACTLY on a focus point.
 it is a teeny, tiny bit off. But that is just a technical detail)
 
 If the earth and the sun were the right size in this drawing, 
 you would need a microscope to see them.
 Also, the earth's ellipse is not stretched out nearly this much.
 
 OK, so since the focus points have real world value,
 it would be good if we were able to calculate where they were.
 
 Here's the deal.
 Say we have an ellipse equation like this:
 

 

 
 Hey, we've seen this one before ...
 
 It looks like this:
 

 

 
 So the focus points are along the stretched out axis.
 In this case, that's the Y axis.
 
 The trick for finding the distance the focus points are from the center
 works like this:
 
 Take the two denominators from the ellipse equation, 
 square them, and find the difference between them.
 Then take the square root of that.
 
 That's the distance from the center to the focus point.
 
 So, as an equation:

 

 
 So since this ellipse is centered at (0,0),
 The focus points are at (0, 8.660) and (0,-8.660)
 

 

 
 On the last page we met this ellipse:
 

 

 
 Where are the foci? (That's the plural of focus!)
 
 OK, this one is stretched in the Y direction,
 and moved one unit down and two units to the right.
 

 

 
 Since the whole ellipse is moved two units to the right,
 the line that the foci are on is moved two units to the right.
 Since the whole ellipse is moved one unit down,
 the left/right centerline is also moved one unit down.
 The center of the ellipse is at (2, -1)
 
 The equation for the distance to the focus points
 works out the same as before ...
 

 

 
 BUT, that distance is measured from the center point of this ellipse.
 
 So the focus points are at (2, 7.660) and (2, -9.660):
 

 

 
 So when the sign between the X 2 and the Y 2 term is a "+"
 We get a circle or an ellipse.
 What would we get if the sign was a minus?
 

   copyright 2005 Bruce Kirkpatrick

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