Ellipses Answers to Question 1-6

1.      

 

 

          Because  is an axis of symmetry.

 

 

          Thus,

 

 

          .

 

 

In summary we can say that the length of the major axis.

 

 

 

 

 

2.      

 

 

          Recalling that  and cm

 

 

          We can conclude that  and.

 

 

          Thus they are each equal to 5.695 cm.

 

 

 

 

3.                 Recall that 5.695 cm and 4.27 cm.

 

 

Using the Pythagorean Theorem

 

 

.  Plugging in values and solving for,  is approximately 3.767305.

 

 

 

 

 

4.      

 

 

          Recall that,

 

 

          Thus

 

 

         

 

 

          This is commonly written as  where.

 

 

a = half the length of the major axis and b = half the length of the minor axis.

 

 

 

 

 

 

  1.  Using what was derived in problem number four,, and plugging in the values for

 

 

and the foci are located approximately 3.768 cm right of the center of the ellipse and 3.768 cm left of the center of the ellipse.

 

 

 

 

 

 

  1. Students should realize that is easy to locate the foci using a compass only.  A circle should be constructed with  as the center and

 

 

 

 

 

  1. segment as the radius.  The intersections of the circle and segment  are where the foci are located.  See picture below.

 

 

 

 

Return to ellipses main page