12.9 Homework: ellipses

In class, we discussed how to draw an ellipse using a piece of cardboard, two thumbtacks, a string, and a pen or marker. Do this. Bring your drawing in as part of your homework. (Yes, this is a real part of your homework!)

$\frac{(x-2{)}^2}{9}+\frac{y^2}{\left(\frac{1}{4}\right)}=1$

• A

  Is it horizontal or vertical?

• B

  What is the center?

• C

  What is $a$ ?

• D

  What is $b$ ?

• E

  What is $c$ ?

• F

  Graph it.

$\frac{4x^2}{9}+25y^2=1$

This sort of looks like an ellipse in standard form, doesn’t it? It even has a 1 on the right. But it isn’t. Because we have no room in our standard form for that 4 and that 25—for numbers multiplied by the x2 and y2 terms. How can we get rid of them, to get into standard form, while retaining the 1 on the right?

• A

  Rewrite the left-hand term, $\frac{4x^2}{9}$ , by dividing the top and bottom of the fraction by 4. Leave the bottom as a fraction; don't make it a decimal.

• B

  Rewrite the right-hand term, $25y^2$ , by dividing the top and bottom of the fraction by 25. Leave the bottom as a fraction; don't make it a decimal.

• C

  Now, you're in standard form. what is the center?

• D

  How long is the major axis?

• E

  How long is the minor axis?

• F

  What are the coordinated of the two foci?

• G

  Graph it.

$18x^2+\frac{1}{2}{y}^2+108x+5y+170=0$

• A

  Put in standard form.

• B

  Is it horizontal or vertical?

• C

  What is the center?

• D

  How long is the major axis?

• E

  How long is the minor axis?

• F

  What are the coordinates of the two foci?

• G

  Graph it.

The major axis of an ellipse runs from (5,-6) to (5,12). One focus is at (5,-2). Find the equation for the ellipse.

The foci of an ellipse are at (-2,3) and (2,3) and the ellipse contains the origin. Find the equation for the ellipse.

We traditionally say that the Earth is 93 million miles away from the sun. However, if it were always 93 million miles away, that would be a circle (right?). In reality, the Earth travels in an ellipse , with the sun at one focus . According to one Web site I found,

There is a 6% difference in distance between the time when we're closest to the sun (perihelion) and the time when we're farthest from the sun (aphelion). Perihelion occurs on January 3 and at that point, the earth is 91.4 million miles away from the sun. At aphelion, July 4, the earth is 94.5 million miles from the sun.

Write an equation to describe the orbit of the Earth around the sun. Assume that it is centered on the origin and that the major axis is horizontal. (*Why not? There are no axes in space, so you can put them wherever it is most convenient.) Also, work in units of millions of miles —so the numbers you are given are simply 91.4 and 94.5.