Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ

Question

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π.
-$$r = 10 + 3 \sin \theta , r = 10 + 3 \cos \theta$$
A) $\left( 10 + \frac { 3 \sqrt { 2 } } { 2 } , \frac { 3 \pi } { 4 } \right) , \left( 10 - \frac { 3 \sqrt { 2 } } { 2 } , \frac { 7 \pi } { 4 } \right)$
B) $\left( 10 - \frac { 3 \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } \right) , \left( 10 + \frac { 3 \sqrt { 2 } } { 2 } , \frac { 7 \pi } { 4 } \right)$
C) $\left( 10 + \frac { 3 \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } \right) , \left( 10 - \frac { 3 \sqrt { 2 } } { 2 } , \frac { 5 \pi } { 4 } \right)$
D) $\left\{ 10 - \frac { 3 \sqrt { 2 } } { 2 } , \frac { \pi } { 4 } , \left( 10 + \frac { 3 \sqrt { 2 } } { 2 } , \frac { 5 \pi } { 4 } \right) \right.$

Quizplus · Accepted Answer

To find the points of intersection, set the two equations equal to each other: $10 + 3\sin\theta = 10 + 3\cos\theta$. Simplifying, we get $\sin\theta = \cos\theta$, which occurs at $\theta = \frac{\pi}{4}$ and $\theta = \frac{5\pi}{4}$ within the given domain. Substituting these values back into either original equation gives the corresponding $r$ values, leading to the points $\left( 10 + \frac{3\sqrt{2}}{2}, \frac{\pi}{4} \right)$ and $\left( 10 - \frac{3\sqrt{2}}{2}, \frac{5\pi}{4} \right)$.

Find the Polar Coordinates of the Point(s) of Intersection of the Given