Chapter 10, Problem 64RE

To determine

Tofind:the rectangular equation and graph the curve using parametric equation.

Answer to Problem 64RE

The rectangular equation is ${\left(\text{y-5}\right)}^{\text{2}}\text{=}\frac{1}{2}\left(\text{x-6}\right)$ and the graph here will be parabolic.

Explanation of Solution

Given:

  ${\text{x=2t}}^2\text{+6, y=5-t}$ ; $\text{-}\infty \text{<t<}\infty$ .

Concept used:

The parametric equation has equation for each variable whereas rectangular equation has one equation

In rectangular equation it composed of variable like $\text{x and y}$ which can be graphed on a regular cartesian plane.

To convert parametric to rectangular $\text{t}$ should be eliminate so that function will be in the form of $\text{x and y}$ .consist of only one independent variable.

Calculation:

According to the given parametric equation:

  ${\text{x=2t}}^2\text{+6}$ …………… $\left(\text{i}\right)$

  $\text{y=5-t}$ …………….. $\left(\text{ii}\right)$

Squaring the equation $\left(\text{ii}\right)$

  ${\text{t}}^{\text{2}}\text{=}{\left(\text{5-y}\right)}^{\text{2}}$ ……….. $\left(\text{iii}\right)$

By putting the value of ${\text{t}}^2$ in equation $\left(\text{i}\right)$ :

  $\text{x=2}{\left(\text{5-y}\right)}^{\text{2}}\text{+6}$ .

  $\text{x-6=2}{\left(\text{5-y}\right)}^{\text{2}}$ .

Which can be written as:

  $2{\left(\text{y-5}\right)}^{\text{2}}\text{=x-6}$ .

  ${\left(\text{y-5}\right)}^{\text{2}}\text{=}\frac{1}{2}\left(\text{x-6}\right)$ .

The equation is of parabola.

Which is rectangular equation.

Since, the power or the degree of the equation is two so the graph will be curve line.

The graph of the equation ${\left(\text{y-5}\right)}^{\text{2}}\text{=}\frac{1}{2}\left(\text{x-6}\right)$ :

  

Here from the graph:

The graph shifted to right at a distance of $\text{x=6}$ and the coordinate point is $\left(6,5\right)$ .

The graph lies in the positive region it shows that domain and range of the given parabolic equation is positive and the function is even function.

Hence, the rectangular equation is ${\left(\text{y-5}\right)}^{\text{2}}\text{=}\frac{1}{2}\left(\text{x-6}\right)$ and the graph here will be parabolic.