Chapter 8.4, Problem 9E

(a)

To determine

To calculate: To sketch the curve represented by the parametric equations

Explanation of Solution

Given information: Parametric equations $x=\frac{1}{t},y=t+1$

Calculation:

Parametric equations are given as

  $x=\frac{1}{t},y=t+1$

Find the value of $x$ and $y$ for values of $t$

When $t=1$

  $\begin{array}{l}x=\frac{1}{1}=1\\ y=1+1=2\end{array}$

When $t=2$

  $\begin{array}{l}x=\frac{1}{2}=0.5\\ y=2+1=3\end{array}$

When $t=4$

  $\begin{array}{l}x=\frac{1}{4}=0.25\\ y=4+1=5\end{array}$

Plotting the point on graph,

  

Conclusion:

Hence, Curve is sketched

(b)

To determine

To calculate: To find the rectangular coordinate equation for the curve

Answer to Problem 9E

The rectangular coordinate equation for the curve is $x\left(y-1\right)=1$

Explanation of Solution

Given information: Parametric equations $x=\frac{1}{t},y=t+1$

Calculation:

Parametric equations are given as

  $x=\frac{1}{t},y=t+1$

From equation $x=\frac{1}{t}$

  $t=\frac{1}{x}---(1)$

From equation $y=t+1$

  $t=y-1--(2)$

Equating (1) and (2)

  $\begin{array}{l}\frac{1}{x}=y-1\\ 1=x\left(y-1\right)\\ x\left(y-1\right)=1\end{array}$

Conclusion:

Hence, rectangular coordinate equation for the curve is $x\left(y-1\right)=1$