Chapter 10, Problem 69RE

To determine

To find two different parametric equations for rectangular equation.

Answer to Problem 69RE

The two parametric equations are $\boxed{x=t}$ and $\boxed{y=-2t+4,-\alpha <t<\alpha }$ ; $\boxed{x=\frac{t-4}{-2}}$ and $\boxed{y=t,-\alpha <t<\alpha }$ .

Explanation of Solution

Given information:

Find two different parametric equations for the given rectangular equation.

Calculation:

Let us assume, $x=t$ .

Now, substitute $t$ for $x$ in the equation $y=-2x+4$ to get $y$ .

  $y=-2t+4$

Hence, the parametric equations are $\boxed{x=t}$ and $\boxed{y=-2t+4,-\alpha <t<\alpha }$ .

Now, we will take another case where, $x=\frac{t-4}{-2}$ .

Substitute $\frac{t-4}{-2}$ for $x$ in the equation $y=-2x+4$ to get $y$ .

  $y=-2\left(\frac{t-4}{-2}\right)+4$

  $y=t-4+4$

  $y=t$

Hence, another set of parametric equations are $\boxed{x=\frac{t-4}{-2}}$ and $\boxed{y=t,-\alpha <t<\alpha }$ .