Describe the graph of the polar equation.
-$$5 r \cos \theta + r \sin \theta = 7$$
A) Vertical line passing through (5, 0)
B) Parabola with vertex (5, 7) opening upward
C) Line with slope -5 and y-intercept (0, 7)
D) Line with slope 7 and y-intercept (0, 5)

Question

Quizplus · Accepted Answer

Rewrite the equation as $r = \frac{7}{5\cos 	heta + \sin 	heta}.$ This indicates that $r$ is always positive, so the graph is a curve that starts at the pole and extends outwards. To determine the shape of the curve, consider the behavior as $	heta$ approaches $0$ and $\frac{\pi}{2}.$ As $	heta$ approaches $0,$ $\cos 	heta \approx 1$ and $\sin 	heta \approx 	heta,$ so $$r \approx \frac{7}{5 + 	heta}.$$As $	heta$ approaches $\frac{\pi}{2},$ $\cos 	heta \approx 0$ and $\sin 	heta \approx 1,$ so $$r \approx \frac{7}{\sin 	heta}.$$Thus, $r$ approaches infinity as $	heta ightarrow 0$ and $	heta ightarrow \frac{\pi}{2}.$ Therefore, the graph is an asymptote (i.e. a line that the curve approaches as $r ightarrow \infty$) with slope $-\frac{5}{1} = -5.$ Since the curve approaches this line from above, the final graph is a line with slope $-5$ and $y$-intercept $(0,7).$

Describe the Graph of the Polar Equation 5rcos⁡θ+rsin⁡θ=75 r \cos \theta + r \sin \theta = 75rcosθ+rsinθ=7

Describe the Graph of the Polar Equation $5 r \cos \theta + r \sin \theta = 7$