Find the polar equation that has an eccentricity of 1 and has a horizontal directrix located 2 units above the pole.
A: r=2/1+sin theta
B: r=2/1+cos theta
C: r=2/1-cos theta
D: r=2/1-sin theta
AAAaannndd the bot gets it wrong yet again!
clearly the vertex lies on the x-axis, and it opens left, not up.
To find the polar equation with an eccentricity of 1 and a horizontal directrix located 2 units above the pole, we can use the fact that the eccentricity of a polar equation is given by e = d/r, where d is the distance from the pole to the directrix and r is the distance from the pole to a point on the curve.
In this case, the eccentricity is given as 1 and the distance from the pole to the directrix is 2 units above the pole, so d = -2.
We also know that the equation has a horizontal directrix, which means that the distance from the pole to a point on the curve can be represented as r = (d + e |sinθ|), where θ is the angle measured counterclockwise from the positive x-axis.
Plugging in the values we have, we get r = (-2 + 1 |sinθ|).
Simplifying this equation, we have r = (-1 |sinθ|).
Comparing this equation with the given options, the correct answer is d: r=2/1-sin theta.
To find the polar equation of a conic section, we need to identify the shape of the conic section based on its eccentricity and the location of its directrix.
In this case, the eccentricity is given as 1, which indicates that the conic section is a parabola.
For a parabola with a horizontal directrix located above the pole, we can determine the polar equation by using the formula:
r = (2 * d) / (1 + e * cos(theta))
Where:
- r is the distance from the pole to a point on the parabolic curve.
- d is the distance between the pole and the directrix.
- e is the eccentricity.
- theta is the angle between the polar axis and the line connecting the pole and the point.
Given that the directrix is located 2 units above the pole, we substitute d = 2 into the equation:
r = (2 * 2) / (1 + e * cos(theta))
Since the eccentricity e is given as 1, we can simplify the equation:
r = 4 / (1 + cos(theta))
Comparing this equation with the answer options:
A: r = 2 / (1 + sin(theta))
B: r = 2 / (1 + cos(theta))
C: r = 2 / (1 - cos(theta))
D: r = 2 / (1 - sin(theta))
We can see that the correct answer is B: r = 2 / (1 + cos(theta)).