Chapter 8.1, Problem 67E

To determine

To convert the given polar equation to rectangular coordinates

  $\sec \theta =2$

Answer to Problem 67E

The rectangular equation is $2x=\sqrt{\left(x^2+y^2\right)}$ .

Explanation of Solution

Given: Polar equation: $\sec \theta =2$

Formula Used:

A polar equation is any equation that describes a relation between r and θ , where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

  $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

  $\sec \theta =\frac{1}{\cos \theta }$

Calculation:

Given Polar equation is $\sec \theta =2$

  $\sec \theta =2$ ;

Solving further, we have:

  $\frac{1}{\cos \theta }=2$

  $2\cos \theta =1$

Multiplying by $r$ , we have:

  $2\left(r\cos \theta \right)=r$

Using the above definition, we have:

  $2x=\sqrt{\left(x^2+y^2\right)}$

Thus, the rectangular equation is $2x=\sqrt{\left(x^2+y^2\right)}$ .

Conclusion:

The rectangular equation is $2x=\sqrt{\left(x^2+y^2\right)}$ .