Question

There are two fruit trees located at (3,0) and (–3, 0) in the backyard plan. Maurice wants to use these two fruit trees as the focal points for an elliptical flowerbed. Johanna wants to use these two fruit trees as the focal points for some hyperbolic flowerbeds. Create the location of two vertices on the y-axis. Show your work creating the equations for both the horizontal ellipse and the horizontal hyperbola. Include the graph of both equations and the focal points on the same coordinate plane.

Answer 1

Maurice wants to create a set of elliptical flower beds. To do this, he first plots the location of the two fruit trees on his graph.

Maurice has to use the equation a^2-b^2=c^2. We know that c=3, and because we need 1 more number to solve for b, I made a=6. 6^2-b^2=3^2. 36-b^2=9. b^2=27. b=5.196

Next, to create the equation, we substitute what we know into the equation x^2/a^2 + y^2/b^2=1 and get x^2/36 + y^2/27=1. Johanna wants to create some hyperbolic flower beds.

We already know that c=3 so this time I decided a=1. 3^2=1^2+b^2. 9=1+b^2. 8=b^2. b=2.828
 
Next, to create the equation, we substitute what we know to the equation x^2/a^2 - y^2/b^2 = 1. x^2/1^2 - y^2/2.828^2 = 1.