Find a polar equation of this conic with focus at the pole and the given eccentricity and directrix.

e=2, r cos X =-1

To find the polar equation of a conic with focus at the pole and the given eccentricity and directrix, we need to first determine the polar coordinates of the focus and the equation of the directrix.

Given: e = 2 and r cos θ = -1.

The eccentricity, e, is defined as the ratio of the distance between the focus and a point on the conic to the distance between the directrix and that same point. In this case, since the focus is at the pole (0, 0), the distance between the focus and a point on the conic is equal to the distance between that point and the origin. Thus, the eccentricity can be calculated as follows:

e = r / (-r cos θ),
2 = -1 / cos θ.

Simplifying, we get:

2cos θ = -1.

Now, we can solve this equation to find the value of θ.

Dividing both sides by 2:

cos θ = -1/2.

The directrix is given by the equation r = -k / (1 + e), where k is the distance from the focus to the directrix. Since the focus is at the pole, the distance k is equal to the absolute value of the constant term in the polar equation. In this case, k = 1.

Using the equation of the directrix, we can substitute the values of e and k to find the equation:

r = -1 / (1 + 2),
r = -1 / 3.

So, the polar equation of the conic with focus at the pole and eccentricity e = 2, and the directrix r cos θ = -1, is:

r = -1/3.

To find a polar equation of a conic with the focus at the pole (origin) and given eccentricity and directrix, you can start by understanding the geometric definition of these elements in a conic section.

In polar coordinates, a conic section can be defined by the equation:

r = 𝑟0 / (1 + ε cos(θ − θ0))

Where:
- r represents the distance from the origin (pole) to a point on the conic section,
- 𝑟0 is the distance from the origin to the directrix (perpendicular to the polar axis),
- ε is the eccentricity of the conic section,
- θ is the angle formed by a point on the conic section and the positive x-axis, and
- θ0 is the angle that the directrix makes with the positive x-axis.

Given that the focus is at the pole and the directrix equation is r cos(θ) = -1, you know that the distance from the origin to the directrix (𝑟0) is 1.

To find the eccentricity (ε), you can use the relation:

ε = distance between the pole and the focus / distance between the pole and the directrix

In this case, the distance between the pole and the focus is 0 (since the focus is at the pole). The distance between the pole and the directrix is 1. Therefore, the eccentricity ε is 0/1 = 0.

Now, substitute the values of 𝑟0 and ε into the equation:

r = 1 / (1 + 0 cos(θ - θ0))

Notice that since ε is 0, the term involving cos(θ - θ0) cancels out.

The resulting polar equation is:

r = 1

Therefore, the polar equation of the given conic with the focus at the pole and the directrix r cos(θ) = -1 is r = 1.