Question

Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.

Accepted Answer

Recognize that the curve is given in polar form as $$r = 8 \cos 	heta - 4. To $$find the slopes of tangent lines at the origin, we need to find the values of $$	heta$$ where the curve passes through the origin, i.e., where $$r = 0. $$Set $$r = 0$$ and solve for $$	heta$$: $$0 = 8 \cos 	heta - 4. $$Rearranging gives $$8 \cos 	heta = 4$$, so $$\cos 	heta = \frac{1}{2}. $$Find the values of $$	heta$$ that satisfy $$\cos 	heta = \frac{1}{2}. $$These are $$	heta = \frac{\pi}{3}$$ and $$	heta = \frac{5\pi}{3} ($$within one full rotation $$0 \leq 	heta \u003c 2\pi). $$Recall that the slope of the tangent line to a polar curve at a point where $$r 
eq 0 is $$given by $$\frac{dy}{dx} = \frac{r' \sin 	heta + r \cos 	heta}{r' \cos 	heta - r \sin 	heta}$$, where $$r' = \frac{dr}{d	heta}. $$However, since the problem asks to avoid calculating derivatives, use the geometric fact that at the origin, the tangent lines correspond to the lines passing through the origin at angles $$	heta$$ where $$r=0. $$Therefore, the slopes of the tangent lines at the origin are the slopes of lines making angles $$	heta = \frac{\pi}{3}$$ and $$	heta = \frac{5\pi}{3}$$ with the positive x-axis. Use $$	ext{slope} = 	an 	heta to $$find the slopes of these tangent lines.

Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.

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Key Concepts

Polar Coordinates and the Curve Equation

Tangent Lines in Polar Coordinates

Slope of Tangent Lines Without Derivatives