$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$
-$\sqrt { 2 } x - \sqrt { 2 } y = 10$
A) $r \cos \left( \theta + \frac { \pi } { 3 } \right) = 10$
B) $r \cos \left( \theta + \frac { \pi } { 4 } \right) = 5$
C) $r \cos \left( \theta + \frac { 3 \pi } { 4 } \right) = 10$
D) $r \cos \left( \theta - \frac { \pi } { 4 } \right) = 5$

Question

Quizplus · Accepted Answer

To convert the given line equation $\sqrt{2}x - \sqrt{2}y = 10$ to polar form, we use the polar coordinates conversion formulas $x = r\cos(\theta)$ and $y = r\sin(\theta)$. Substituting these into the given equation yields $\sqrt{2}r\cos(\theta) - \sqrt{2}r\sin(\theta) = 10$. Factoring out $\sqrt{2}r$ gives $\sqrt{2}r(\cos(\theta) - \sin(\theta)) = 10$. Using the angle sum identity, we recognize that $\cos(\theta) - \sin(\theta)$ can be rewritten in the form of a cosine function of a shifted angle, specifically $\cos(\theta - \frac{\pi}{4})$, because $\cos(\theta - \frac{\pi}{4}) = \cos(\theta)\cos(\frac{\pi}{4}) + \sin(\theta)\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}\cos(\theta) + \frac{\sqrt{2}}{2}\sin(\theta)$, which simplifies to $\cos(\theta) - \sin(\theta)$ when multiplied by $\sqrt{2}$. Therefore, the equation becomes $\sqrt{2}r\cos(\theta - \frac{\pi}{4}) = 10$, and dividing both sides by $\sqrt{2}$ gives $r\cos(\theta - \frac{\pi}{4}) = 5$, which matches choice B.

Find a polar equation in the form rcos⁡(θ−θ0)=r0 for the given line. \text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. } Find a polar equation in the form rcos(θ−θ0​)=r0​ for the given line.

$\text { Find a polar equation in the form } r \cos \left( \theta - \theta _ { 0 } \right) = r _ { 0 } \text { for the given line. }$