Question

for each of the described curves, decide if the curve would be more easily given by a polar equation or a cartesian equation. then write an equation for the curve. (a) a line through the origin that makes an angle of π/6 with the positive x - axis. (b) a vertical line through the point (9,9).

Explanation:

Step1: Determine the type of equation for part (a)

A line through the origin with an angle of \(\frac{\pi}{6}\) with the positive \(x -\)axis is more easily given by a polar equation. In polar coordinates, the equation of a line passing through the origin (pole) with an angle \(\theta=\alpha\) with the positive \(x -\)axis is \(\theta=\alpha\). Here \(\alpha = \frac{\pi}{6}\), so the polar equation is \(\theta=\frac{\pi}{6}\)

Step2: Determine the type of equation for part (b)

A vertical line \(x = c\) (in Cartesian coordinates) where \(c\) is a constant. A vertical line through the point \((9,9)\) has the Cartesian equation \(x = 9\). In polar coordinates, \(x=r\cos\theta\). If we write \(x = 9\) in polar form, we get \(r\cos\theta=9\) or \(r=\frac{9}{\cos\theta}\). But a vertical line is more straightforwardly represented in Cartesian form as \(x = 9\)

Answer:

(a) Polar equation, \(\theta=\frac{\pi}{6}\)
(b) Cartesian equation, \(x = 9\)