Chapter 9.4, Problem 44E

To determine

To calculate:

The set of parametric equations for the given conic section:

Conic section: hyperbola

Vertices: $\left(\pm 2,0\right)$

Foci: $\left(\pm 3,0\right)$

Answer to Problem 44E

The set of parametric equations for hyperbola is $\left(2\sec \theta ,2\sqrt{3}\tan \theta \right)$ .

Explanation of Solution

Given information:

Conic section: hyperbola

Vertices: $\left(\pm 2,0\right)$

Foci: $\left(\pm 3,0\right)$

Formula used:

The parametric equations of hyperbola conic section are given by the following expression:

  $\begin{array}{l}x=h+a\sec \theta \\ y=k+b\tan \theta \end{array}$

Calculation:

The given hyperbola has vertices $\left(\pm 2,0\right)$ , so: $h=0,k=0$ .

Now, the length of major axis is $a=2$

The foci is given as $\left(\pm 3,0\right)$ , so $c=3$

Now, need to find the value of $b$ by using the formula:

  $\begin{array}{l}{b}^2=c^2-a^2\\ b^2=4^2-2^2\\ b^2=12\\ b=2\sqrt{3}\end{array}$

The parametric equations of hyperbola conic section are given by the following expression:

  $\begin{array}{l}x=h+a\sec \theta \\ y=k+b\tan \theta \end{array}$

Put the values:

  $\begin{array}{l}x=h+a\sec \theta =0+2\sec \theta =2\sec \theta \\ y=k+b\tan \theta =0+2\sqrt{3}\tan \theta =2\sqrt{3}\tan \theta \end{array}$