The distance between the foci will be the difference between the point on the orbit furthest from one of the foci and the point on the orbit nearest to the same foci.
B
b
The distance between two foci is equal to 2c. Use the answer from Part A to solve for c.
C
c
Use the fact c^2=a^2-b^2 to solve for b^2.
A
a
3 * 10^6 miles
B
b
About 0.016
C
c
x^2/8.649 * 10^(15)+y^2/8.64675 * 10^(15)=1
Practice makes perfect
a
We know that the sun is at one of the foci, (± c,0). Let's assume the sun is at the focus (- c,0). We are told the distance between the point on the elliptical nearest to the sun and the sun itself is 9.15 * 10^7 miles. Similarly, we know the point furthest from the sun is 9.45 * 10^7 miles away. Let's label these distances.
The major axis contains the foci and has vertices on the ellipse. The vertex (- a,0) is 9.15 * 10^7 miles from the sun. The other vertex (a,0) is 9.45 * 10^7 miles from the sun. The distance between the foci will be the difference between the point on the orbit furthest from a focus and the point on the orbit nearest to the same focus. Let's solve for this!
The distance between the sun and the other focus is 3 * 10^6 miles.
b
We are asked to find the eccentricity of the orbit. Let's first recall the equation for eccentricity.
Eccentricity=c/a
Here, c is the distance from the center to a focus, and a is the distance from the center to a vertex. Let's start by finding c. The distance between two foci is equal to 2c. Using the answer from Part A, we can solve for c.
Now we can solve for a. The length of a major axis is equal to 2a. Looking at our elliptical drawn in Part A, we can see that the major axis is equal to the distance from the sun to the point on the orbit closest to it plus the distance from the sun to the point on the orbit furthest from it.
Let's recall the form for the standard equation of a horizontal ellipse.
x^2/a^2+y^2/b^2=1In order to find b^2, we will use the fact c^2=a^2-b^2 for an ellipse. Let's substitute the values for c and a that we found in Part B.
We have found b^2=8.64675 * 10^(15). In the process of solving for b^2, we also found the value of a^2=86.49 * 10^(14), or 8.649 * 10^(15) in scientific notation. Finally, let's substitute these values into the standard equation of a horizontal ellipse.
x^2/a^2+y^2/b^2=1
⇕
x^2/8.649 * 10^(15)+y^2/8.64675 * 10^(15)=1
