Question

Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.

Accepted Answer

Recall that the polar curve is given by $$r = 4 \cos 	heta. $$The point at the origin corresponds to $$r = 0. $$Find the values of $$	heta$$ where $$r = 0 by $$solving $$4 \cos 	heta = 0. $$Identify the points on the curve where it passes through the origin. These occur at the values of $$	heta$$ found in the previous step. For each such $$	heta$$, the curve passes through the origin $$(r=0). To $$find the slope of the tangent line in Cartesian coordinates, convert the polar coordinates to Cartesian using $$x = r \cos 	heta$$ and $$y = r \sin 	heta. $$Then express $$y as a $$function of $$x$$ implicitly or find $$\frac{dy}{dx}$$ using the chain rule and derivatives with respect to $$	heta. $$Use the formula for the slope of the tangent line to a polar curve: $$\frac{dy}{dx} = \frac{\frac{dr}{d	heta} \sin 	heta + r \cos 	heta}{\frac{dr}{d	heta} \cos 	heta - r \sin 	heta}. $$Compute $$\frac{dr}{d	heta}$$ and substitute $$r$$, $$	heta$$, and $$\frac{dr}{d	heta} at $$the point where $$r=0. $$Evaluate the slope at the origin. If the denominator of $$\frac{dy}{dx} is $$zero while the numerator is nonzero, the slope is undefined, indicating a vertical tangent line. This explains why the slope of the tangent line at the origin is undefined.

Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.

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Key Concepts

Polar Coordinates and Polar Curves

Tangent Lines in Polar Coordinates

Undefined Slope and Vertical Tangents