Chapter H.1, Problem 21E

(a)

To determine

To decide: whether the given condition is easy to draw a polar equation or a Cartesian equation and write an equation for the curve.

Answer to Problem 21E

The Cartesian equation of the line is $y=(\tan (\pi /6))x$

It is more difficult to give the Cartesian equation for the curve.

Explanation of Solution

Given:

A line through the origin that makes an angle of $\pi /6$ with the positive $x$ -axisCalculation:

The curve would be more easily given by a polar equation $[9=\pi /6]$ For the Cartesian equation the slope of the line is $=\tan (\pi /6)$ Then Cartesian equation of the line is $y=(\tan (\pi /6))x$

Or $y=\frac{1}{\sqrt{3}}x$

It is more difficult to give the Cartesian equation for the curve.

Conclusion:

Therefore, theCartesian equation of the line is $y=(\tan (\pi /6))x$

It is more difficult to give the Cartesian equation for the curve.

(b)

To determine

To decide: whether the given condition is easy to draw a polar equation or a Cartesian equation and write an equation for the curve.

Answer to Problem 21E

The curve would be more easily given by Cartesian equation. Equation of the curve is $[x=3]$

Explanation of Solution

Given:

A vertical line through the point (3.3)

Calculation:

The curve would be more easily given by Cartesian equation. Equation of the curve is $[x=3]$

Conclusion:

The curve would be more easily given by Cartesian equation. Equation of the curve is $[x=3]$