Moons Orbit Ellipse: Perilune, Apolune, Coordinates

[js14]

For an object in an elliptical orbit around the moon, the points in the orbit that are closest to and farthest from the center of the moon are called perilune and apolune, respectively. These are the vertices of the orbit. The center of the moon is at one focus of the orbit. A spacecraft was placed in a lunar orbit with perilune at a = 71 mi and apolune at b = 204 mi above the surface of the moon. Assuming that the moon is a sphere of radius 1075 mi, find an equation for the orbit of this spacecraft . (Place the coordinate axes so that the origin is at the center of the orbit and the foci are located on the x-axis. Round each answer to the nearest whole number.)

I really have no idea of how to solve this. Can someone help?

 

[praharmitra]

Do you know the equation of the ellipse? x^2/m^2 + y^2/n^2 = 1. Your job simply is to calculate m and n.

First draw a diagram and mark all the lengths that are given to you. (Eg - Perilune length = 71+1075). It should be easy to see then.

 

[js14]

im still a little confused... so I am supposed to add 71 to 1075 and add 204 to 1075?

 

[praharmitra]

Ya, because perilune = 71 mi above the surface of the moon = 71 + distance from center to surface of moon = 71 + 1075. Similarly for apolune. But these give you only a and b. Using these values you need to find m and n.

First draw an ellipse and identify which points correspond to distances a and b, from the focus (farthest and smallest distance).

 

[js14]

i still don't think I am getting it. I am getting 1143 for a and 1635841 for b.

 

[js14]

if i add 71 to 1075 and 204 to 1075 i get a= 1143 and b= 1270. I don't know what to do next...

 

[praharmitra]

How?!? you are confusing a and b, I believe. The standard equation of an ellipse is

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

These are NOT the same a and b that you are working with! To avoid confusion all together, let us call -

Perilune = closest distance from focus = p = 75 + 1075 (did you understand how that came?)
Apolune = farthest distance from focus = q = 204 + 1075 (same as above)

Did you draw an ellipse like I asked? Now mark one focus (any one, doesn't matter). What is the farthest and closest point to this focus??

 

[js14]

I have a graph that is associated with this problem and it has perolune as the closes. I still am confused though. I know the x^2/a^2 + y^2/b^2 = 1 equation I hadnt ever seen a problem like this until today so I am having trouble setting it set up right. anyway doesn't 71 + 1075 = 1146?

 

[js14]

You act like I am not supposed to add them together?

 

[vela]

I think there's some confusion arising from the notation, so let's get that straightened out first.

In the original post, you said a and b are the heights of, respectively, the perilune and apolune measured from the surface of the moon. One focus of the ellipse lies at the center of the moon, so the distance p from the focus to the perilune is p=a+R, where R is the radius of the moon, and the distance q from the focus to the apolune is q=b+R. So you do want to do those additions.

Now you have the equation of the ellipse

$$\frac{x^2}{m^2} + \frac{y^2}{n^2} = 1$$

which introduces two more distances, m and n. You need to figure out how to find m and n from p and q. To do that, follow praharmitra's advice and draw a sketch of the orbit. You should be able to easily find what m equals from the sketch. With a little more work, you can find n.

 

[js14]

Its cool I finally figured it out. the answer is x^2/1470156 + y^2/1465734 = 1. It really wasnt that difficult I was just a little confused because I had never seen a problem like this and I've only been doing analytic geometry for a couple of days. Thanks for your help though!