# The point in a lunar orbit nearest the surface of the moon is called perilune and the point...

## Question:

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 110 km and apolune altitude 314 km (above the moon). Find an equation of this ellipse if the radius of the moon is 1728 km and the center of the moon is at one focus.

## Ellipse

This problem is based on the application of the equation of the ellipse. We have been given the radius of the moon and we are going to use the formulas to find the major and minor axis.

We will begin by finding the major and minor axis.

{eq}\displaystyle 2a=2(1728)+(110+314) {/eq}

{eq}\displaystyle 2a=2(1728)+(424) {/eq}

{eq}\displaystyle a=1940 {/eq}

Let us now find the focus point,

{eq}\displaystyle 2c=(1728+314)-(1728+110) {/eq}

{eq}\displaystyle 2c=204 {/eq}

{eq}\displaystyle c=102 {/eq}

Now, minor axis can be found as:

{eq}\displaystyle b^2=a^2-c^2 {/eq}

{eq}\displaystyle b^2=(1940)^2-(102)^2 {/eq}

{eq}\displaystyle b^2=3753196 {/eq}

Therefore, the equation of ellipse is:

{eq}\displaystyle \boxed{\displaystyle \frac{x^2}{3763600}+\frac{y^2}{3753196}=1} {/eq} 