Chapter 10.5, Problem 24PPSE

Solution

To find: The information need to write an equation and to use the given information to find the information we need.

Given information:

The path that Voyager $2$ made around Jupiter followed one branch of a hyperbola. Find an equation that models the path of Voyager $2$ around Jupiter, given that $a=2,184,140\text{km}$ and $c=2,904,906.2\text{km}$ . Use the horizontal model.

• What information you need to write the equation $?$
• How can you use the given information to find the information you need $?$

Explanation:

The path that Voyager $2$ made around Jupiter is horizontal hyperbola. The standard equation is:

  $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\mathrm{...}\left(1\right)$

The information we need is to find the value of $b^2$ using the value of $c$ and $a$ :

Now, use $c$ and $a$ to find $b^2$ :

  $\begin{array}{l}{b}^2=c^2-a^2\\ =2,904,{906.2}^2-2,184,{140}^2\\ =8,438,480,030,798-4,770,467,539,600\\ =3,668,012,491,198\end{array}$

Then, use this information to find the equation of hyperbola.

Using the value of $a^2$ and $b^2$ into the $\left(1\right)$ equation:

  $\frac{x^2}{4,770,467,539,600}-\frac{y^2}{3,668,012,491,198}=1$