Question

In Exercises \(27-38,\) find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) \(\frac{\text { Conic }}{\text { Parabola }} \quad \frac{\text { Eccentricity }}{e=1} \quad \frac{\text { Directrix }}{x=1}\)

Short answer

The polar equation for the conic with its focus at the pole is \(r = \frac{1}{1 + \cos(\theta)}\).

Step-by-step solution

Analyze given details

Firstly, the given necessary details are: the conic section is a parabola, the eccentricity (\(e\)) equals 1, and the equation for the directrix is \(x=1\), given in rectangular coordinate form.

Recall the polar equation of a conic

A conic section's polar equation with focus at the pole and directrix \(x = d\) is obtained as \(r = \frac{ed}{1 + e\cos(\theta)}\) where \(e\) is the eccentricity, \(d\) is the distance from the origin to the directrix, and \(r\) and \(\theta\) are the polar coordinates. Since this is a parabola, \(e=1\). The equation becomes \(r = \frac{d}{1 + \cos(\theta)}\).

Substitute the value for directrix into the equation

Since the equation for the directrix is \(x=1\), the distance from the origin to the line \(x=1\) (or the directrix) is \(d=1\). Hence, substituting \(d=1\) back into the polar equation, we obtain \(r = \frac{1}{1 + \cos(\theta)}\). This is the required polar equation for the conic.