Chapter 10, Problem 70RE

To determine

To find two different parametric equations for rectangular equation.

Answer to Problem 70RE

The one set of parametric equations for a given parabola are $\boxed{\left(t,2t^2-8\right)}$ .

The another set of parametric equations for a given parabola are $\boxed{\left(\sqrt{t},2t-8\right)}$ .

Explanation of Solution

Given information:

Find two different parametric equations for rectangular equation.

Calculation:

The given equation is that of a parabola which is symmetrical to the y-axis in rectangular Cartesian coordinates.

  $y=2x^2-8$

The parametric equations can be determined for the given parabola by assuming a relation for $x$ with a parameter $t$ and then substituting that relation in the above given expression to obtain a relation between $y$ and $t$ .

Let us suppose, $x=t$ as a parametric equation in $t$ . We will substitute $x$ in the above given expression to obtain the value of $y$ in terms of $t$ .

  $y=2x^2-8$

  $y=2t^2-8$

Now, one set of parametric equations for a given parabola are $\boxed{\left(t,2t^2-8\right)}.$

The another set can be obtained by keeping $x=\sqrt{t}$ and obtaining the value of $y$ in terms of $t$ , which is $y=2t-8$ .

Hence, the another set of parametric equations are $\boxed{\left(\sqrt{t},2t-8\right)}.$