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.) \(\frac{\text { Conic }}{\text { Ellipse }} \quad \frac{\text { Eccentricity }}{e=\frac{1}{2}} \quad \frac{\text { Directrix }}{x=1}\)

Short answer

The polar equation for the ellipse is \(r=\frac{1}{2 - cos(\theta)}\).

Step-by-step solution

Substitute the value of e into the formula

Firstly, substitute e=0.5 into the equation to get \(r=\frac{0.5d}{1 - 0.5cos(\theta)}\).

Find the value of d

The equation of the directrix is x=1. The perpendicular distance, d, from the directrix to the pole (the origin) is the absolute value of the x-coordinate of the directrix which is |1|=1. Therefore, d=1.

Substitute the value of d into the equation

Then substitute d=1 into the equation. This gives \(r=\frac{0.5 × 1}{1 - 0.5cos(\theta)} = \frac{0.5}{1 - 0.5cos(\theta)}\) which simplifies to \(r=\frac{1}{2 - cos(\theta)}\).