Question

Give the equation in polar coordinates of a conic section with a focus at the origin, eccentricity \(e,\) and a directrix \(x=d,\) where \(d>0\)

Short answer

Answer: The polar equation of the conic section is given by \(r = e |r\cos{\theta} - d|\).

Step-by-step solution

Express the distance between a point in the conic section and the directrix

Using the Cartesian coordinates, the directrix is a vertical line given by x=d. Let a point P in the conic section be represented by \((x,y)\) or \((r,\theta)\) in Cartesian and polar coordinates, respectively. Then the distance between the directrix and P is given by \(|x-d|\). We need to express this distance in polar coordinates. Since \(x = r\cos{\theta}\), we can rewrite the distance as \(|r\cos{\theta} - d|\).

Express the distance between a point in the conic section and the focus

The focus of the conic section is at the origin, and we need to express the distance between a point P in the conic section and the origin in polar coordinates. This distance is simply the radial distance r.

Relate the distances using the definition of eccentricity

The definition of the eccentricity e of a conic section is the ratio of the distance between a point P and the focus to the distance between P and the directrix. In our case, this is given by: \[e = \frac{r}{|r\cos{\theta} - d|}\]

Isolate r to express the polar equation

To find the polar equation of the conic section, we need to isolate r from the equation above. We can do this by multiplying both sides by \(|r\cos{\theta} - d|\) and then rearranging the terms: \[r = e |r\cos{\theta} - d|\] This is the polar equation of the conic section with a focus at the origin, eccentricity e, and directrix x=d.