Question

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).
e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).

Accepted Answer

Recall that the polar coordinates $$(r, 	heta)$$ relate to Cartesian coordinates $$(x, y) by $$the equations $$x = r \cos 	heta$$ and $$y = r \sin 	heta$$, where $$r = f(	heta)$$ for the given curve. To find the slope of the tangent line $$\ell at $$point $$P$$, we need to compute $$\frac{dy}{dx}. $$Using the chain rule, express $$\frac{dy}{dx} as$$ $$\frac{\frac{dy}{d	heta}}{\frac{dx}{d	heta}}. $$Calculate $$\frac{dx}{d	heta}$$ and $$\frac{dy}{d	heta}$$ using the product rule:

$$\frac{dx}{d	heta} = f'(	heta) \cos 	heta - f(	heta) \sin 	heta$$,

$$\frac{dy}{d	heta} = f'(	heta) \sin 	heta + f(	heta) \cos 	heta. $$Substitute these derivatives into the expression for the slope of the tangent line:

$$\frac{dy}{dx} = \frac{f'(	heta) \sin 	heta + f(	heta) \cos 	heta}{f'(	heta) \cos 	heta - f(	heta) \sin 	heta}. $$Since the tangent line $$\ell is $$parallel to the y-axis, its slope is undefined, which means the denominator of $$\frac{dy}{dx}$$ must be zero:

$$f'(	heta) \cos 	heta - f(	heta) \sin 	heta = 0.

$$Rearranging this gives:

$$f'(	heta) \cos 	heta = f(	heta) \sin 	heta$$,

which leads to

$$	an 	heta = \frac{f(	heta)}{f'(	heta)}.

$$This completes the proof.

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).
e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).

Verified video answer for a similar problem:

Key Concepts

Polar Coordinates and Conversion to Cartesian Coordinates

Slope of the Tangent Line to a Polar Curve

Condition for a Line to be Parallel to the y-axis

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).

Verified video answer for a similar problem:

Key Concepts

Polar Coordinates and Conversion to Cartesian Coordinates

Slope of the Tangent Line to a Polar Curve

Condition for a Line to be Parallel to the y-axis

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).
e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).