Find a Cartesian equation for the line whose polar equation is given.
-$\mathrm { r } \cos \left( \theta + \frac { \pi } { 4 } \right) = 5 \sqrt { 2 }$
A) $y = x + 5 \sqrt { 2 }$
B) $y = x - 10$
C) $y = \sqrt { 2 } x - 10$
D) $y = x + 5$

Question

Quizplus · Accepted Answer

The polar equation $r \cos(\theta + \frac{\pi}{4}) = 5\sqrt{2}$ can be converted to Cartesian coordinates using the identities $r\cos\theta = x$ and $r\sin\theta = y$. The given equation can be rewritten using the cosine addition formula: $r(\cos\theta\cos\frac{\pi}{4} - \sin\theta\sin\frac{\pi}{4}) = 5\sqrt{2}$. Since $\cos\frac{\pi}{4} = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$, the equation becomes $r(\frac{\sqrt{2}}{2}x - \frac{\sqrt{2}}{2}y) = 5\sqrt{2}$, which simplifies to $x - y = 10$, or equivalently, $y = x - 10$.

Find a Cartesian Equation for the Line Whose Polar Equation rcos⁡(θ+π4)=52\mathrm { r } \cos \left( \theta + \frac { \pi } { 4 } \right) = 5 \sqrt { 2 }rcos(θ+4π​)=52​

Find a Cartesian Equation for the Line Whose Polar Equation $\mathrm { r } \cos \left( \theta + \frac { \pi } { 4 } \right) = 5 \sqrt { 2 }$