Find equation for the curve

Question

a weather satellite with an orbit about the earth reaches a minimum altitude of 1000 miles and a maximum altitude of 1600 miles. the path of its orbit is approximately an ellipse with the center of the earth at one focus. find the equation of this curve. assume the radius of the earth is 4000 miles and the x-axis is the direction of the major axis.

Answer 1

Placing the origin of our system at a focus located at Earth's center, we find the maximum and minimum altitudes occur as the x-intercepts (since we choose the x-axis as the major axis).

 

The minimum distance from the focus = (radius of Earth)+(minimum altitude) = 4000+1000 = 5000

The maximum distance from the focus = (radius of Earth)+(maximum altitude)=4000+1600 = 5600

 

The sum of these distances is twice that from the center of the ellipse to the vertices, which is a.  So,

a = (5000+5600)/2 = 10600/2 = 5300.

 

To find b, we must first find c and use the equation b² + c² = a².  The distance c is the distance from the center of the ellipse to the focus.  we know that (shorter distance from vertex to focus) + c = a

 

5000 + c = 5300

c = 300.

 

Now we find c:  300² + b² = 5300², or b² = 5300² - 300²,

b=2000√(7).

 

Since the problem doesn't specify the location of the origin relative to the orbit of the satellite, we can assume it at the center of the ellipse, which will save us the hassle of performing a translation.

 

We then have x²/(5300)² + y²/(2000√7)² = 1.

 

Answer 2

The key to the problem is that the altitude is not the same thing as the minor or major axes. Altitude is the distance above the surface of the earth. Therefore, the major and minor axes are the altitudes + earth radius. Put this information into the ellipse equation and you will find your answer.