Chapter H.1, Problem 18E

To determine

To find: The polar equation of the curve

Answer to Problem 18E

The polar equation of the curve is $r=9\left(\cos \theta -sin\theta \right)\sec 2\theta$

Explanation of Solution

Given:

  $x+y=9$

Calculation:

Given Cartesian equation $x+y=9$

We have $x=r\cos \theta$

  $y=r\sin \theta$

  $\Rightarrow x+y=r\left(\cos \theta +\sin \theta \right)$

  $\Rightarrow r\left(\cos \theta +\sin \theta \right)=9$

  $\Rightarrow r=\frac{9}{\cos \theta +\sin \theta }$

  $\Rightarrow r=\frac{9\left(\cos \theta -\sin \theta \right)}{{\cos }^2\theta -{\sin }^2\theta }$

  $\Rightarrow r=\frac{9\left(\cos \theta -\sin \theta \right)}{\cos 2\theta }$

  $\Rightarrow r=9\left(\cos \theta .\sec 2\theta -\sin \theta \sec 2\theta \right)$

  $\therefore$ The polar equation of the curve is

  $r=9\left(\cos \theta -sin\theta \right)\sec 2\theta$

Conclusion:

Therefore, the polar equation of the curve is $r=9\left(\cos \theta -sin\theta \right)\sec 2\theta$ .