Question

Q.04: (11 points) Given the polar curve r = e θ , where 0 ≤ θ ≤ 2π. Find points on the curve in the form (r, θ) where there is a horizontal or vertical tangent line. g

Answer 1

I suppose the curve is $r(\theta)=e^\theta$.

Tangent lines to the curve have slope $\frac{dy}{dx}$; use the chain rule to get this in polar coordinates.

$\dfrac{dy}{dx}=\dfrac{dy}{d\theta}\dfrac{d\theta}{dx}=\dfrac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}$

We have

$y(\theta)=r(\theta)\sin\theta\implies\dfrac{dy}{d\theta}=\dfrac{dr}{d\theta}\sin\theta+r(\theta)\cos\theta$

$x(\theta)=r(\theta)\cos\theta\implies\dfrac{dx}{d\theta}=\dfrac{dr}{d\theta}\cos\theta-r(\theta)\sin\theta$

$r(\theta)=e^\theta\implies\dfrac{dr}{d\theta}=e^\theta$

$\implies\dfrac{dy}{dx}=\dfrac{e^\theta\sin\theta+e^\theta\cos\theta}{e^\theta\cos\theta-e^\theta\sin\theta}=\dfrac{\sin\theta+\cos\theta}{\cos\theta-\sin\theta}$

The tangent line is horizontal when the slope is 0, which happens wherever the numerator vanishes:

$\sin\theta+\cos\theta=0\implies\sin\theta=-\cos\theta\implies\tan\theta=-1$

$\implies\theta=\boxed{-\dfrac\pi4+n\pi}$

(where $n$ is any integer)

The tangent line is vertical when the slope is infinite or undefined, which happens when the denominator is 0:

$\cos\theta-\sin\theta=0\implies\sin\theta=\cos\theta\implies\tan\theta=1$

$\implies\theta=\boxed{\dfrac\pi4+n\pi}$