Find the slope of the polar curve at the indicated point.
-$$r = 5 \cos \theta + 8 \sin \theta , \theta = 0$$
A) $- \frac { 5 } { 8 }$
B) $\frac { 5 } { 8 }$
C) $- \frac { 8 } { 5 }$
D) $\frac { 8 } { 5 }$

Question

Quizplus · Accepted Answer

To find the slope of a polar curve at a given point, we use the formula $\frac{dy}{dx} = \frac{\frac{dy}{d	heta}}{\frac{dx}{d	heta}}$. In this case, $x = r\cos	heta = 5\cos^2	heta + 8\sin	heta\cos	heta$ and $y = r\sin	heta = 5\sin	heta\cos	heta + 8\sin^2	heta$. Differentiating both $x$ and $y$ with respect to $	heta$, we get
\begin{align*}
\frac{dx}{d	heta} &= -10\sin	heta\cos	heta + 8\cos^2	heta - 8\sin^2	heta = -2\sin 2	heta \
\frac{dy}{d	heta} &= 5\cos^2	heta - 10\sin	heta\cos	heta - 16\sin	heta\cos	heta = 5\cos^2	heta - 26\sin	heta\cos	heta.
\end{align*}
Therefore, at $	heta = 0$, we have $\frac{dy}{dx} = \frac{5}{-2\sin 0} = \boxed{\frac{5}{8}}$.

Find the Slope of the Polar Curve at the Indicated r=5cos⁡θ+8sin⁡θ,θ=0r = 5 \cos \theta + 8 \sin \theta , \theta = 0r=5cosθ+8sinθ,θ=0

Find the Slope of the Polar Curve at the Indicated $r = 5 \cos \theta + 8 \sin \theta , \theta = 0$