Question

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=4 \sec \theta, \quad y=3 \tan \theta $$

Short answer

The corresponding rectangular equation is \( x^2 (1 - (\frac{3y}{4})^2 ) = 16 \).

Step-by-step solution

Graph the Parametric Equations

With the provided parametric equations, plot the points by substituting a range of values for \( \theta \). The orientation of the curve is counterclockwise because as \( \theta \) increases, the point \( (x, y) \) moves counterclockwise.

Identify the Relationship between x and y

Looking at the parametric equations, we see that \( x = 4 \sec \theta \) and \( y = 3 \tan \theta \). Expressing \( \sec \) and \( \tan \) in terms favorable for elimination leads to \( x = 4 / \cos \theta \) and \( y = 3 \sin \theta / \cos \theta \). Observe that there is \( \cos \theta \) in both expressions which will enable the possibility of dividing \( x \) by \( y \) or vice versa.

Eliminate the Parameter

Divide the first equation by the second, and simplify. This yields: \( x/y = (4 / \cos \theta) / (3 \sin \theta / \cos \theta) \). Simplifying this yields \( x/y = 4/3 \sin \theta \). As we know that \( \sin^2 \theta + \cos^2 \theta = 1 \) , we can write it in terms of \( \cos \theta \) to get \( \cos^2 \theta = 1 - (\frac{3y}{4})^2 \) . Substituting this into \( x \), we obtain: \( x = 4 / \sqrt{1 - (\frac{3y}{4})^2} \). Squaring both sides yields \( x^2 = 16 / ( 1- (\frac{3y}{4})^2) \). This simplifies to the rectangular equation \( x^2 (1 - (\frac{3y}{4})^2 ) = 16 \).