# In your studies, you learned how to write the equation for an ellipse.

August 30, 2017

estion
In your studies, you learned how to write the equation for an ellipse. The equation for an ellipse depends on the orientation of its major axis. The equation for an ellipse with a horizontal major axis is x^2/a^2+y^2/b^2=1 , where a and b are the coordinates of the endpoints of the major and minor axes, respectively, and the ellipse is centered at the origin.

There are many real-world applications for the equation of an ellipse. One of the most important uses is to model planetary motion.

Assignment: Ellipse Application
The planets in our solar system do not travel in circular orbits. Rather, their orbits are elliptical in
shape with the Sun located at one of the foci of the ellipse. The major axis of the elliptical orbit
lies along the x-axis.
The perihelion is the closest a planet comes to the Sun.
The aphelion is the planet’s furthest distance from the Sun.
A planet’s mean distance from the Sun is equal to one-half of the length of the major axis of the
ellipse, as shown in the figure below.
In the sketch below, elliptical orbit is exaggerated to demonstrate the difference between the
perihelion and the aphelion. In reality, the orbit is much more circular.

mean distance

Perihelion

SUN

Aphelion

major axis
center

Think about the elliptical orbit of the planet Mercury. The closest that Mercury comes to the Sun is
about 46 million miles, while the farthest away is about 70 million miles.
Use this information and the standard equation for an ellipse to solve each problem. Show all
For all calculations use “million miles.” For example, write “46 million miles” rather than
“46,000,000.”

1. What are the aphelion and perihelion of Mercury? Explain what each of these measurements
means.

2. The sum of the aphelion and perihelion is equal to the length of the major axis of the elliptical
orbit. Find the approximate length of Mercury’s major axis.

3. Use the length you calculated for the major axis to find Mercury’s mean distance from the Sun.

x2 y 2
+
= 1 . The orbit for Mercury
a2 b2
can be plotted on the coordinate plane with the center at the origin and the Sun located at one
focus. Use this information and the Mercury’s calculated mean distance from the Sun to find the
4. The equation for an ellipse with a horizontal major axis is

5. To write an equation for the elliptical orbit of Mercury, it is necessary to find the value of b. The
equation b 2 = a 2 − c 2 can be used to find the value of b. Find the value of b rounded to the
nearest million miles. (Hint: c is equal to the coordinate of the focus, which is the location of the
Sun. To find the Sun’s location, subtract Mercury’s perihelion from its mean distance from the