What's the polar equation of an ellipse given the eccentricity=1/2 directrix=y=? also how do i know if it is sin or cosine?
recall that the ellipse with eccentricity e and directrix x=d is
r = ed/(1 + e*cosθ)
If you have a y=d directrix then it uses sinθ
To find the polar equation of an ellipse with a given eccentricity and directrix, we can use the definition of an ellipse in terms of its focus, directrix, and eccentricity.
Step 1: Determine the coordinates of the focus.
Since the eccentricity is given as 1/2, we know that the distance from the center to each focus is half the distance of the major axis. Let's assume the major axis is horizontal (along the x-axis). Since the eccentricity is less than 1, we can also conclude that the major axis lies along the x-axis and the minor axis lies along the y-axis.
Let the distance from the center to each focus be a. The distance from the center to the directrix (in this case, y = d) is also a.
Since the eccentricity is 1/2, we can write the following equation:
e = √(1 - (b^2/a^2)) = 1/2
Here, b represents the distance from the center to each vertex (half the length of the minor axis).
From this equation, we can solve for b^2 in terms of a^2:
1/4 = 1 - (b^2/a^2)
Rearranging the equation, we have:
b^2 = a^2 - (a^2)/4
b^2 = (3a^2)/4
Step 2: Write the polar equation.
The polar equation of an ellipse with the focus at the origin, the major axis along the x-axis, and the directrix given by y = d can be expressed as:
r = (2 * d * (1 - e^2)) / (1 - e * cos(θ))
In our case, since the directrix is given by y = d, we have d = a. We already found b^2 = (3a^2)/4.
Substituting these values into the polar equation, we get:
r = (2 * a * (1 - (1/4))) / (1 - (1/2) * cos(θ))
r = (2a * (3/4)) / (1 - (1/2) * cos(θ))
r = (3a/2) / (1 - (1/2) * cos(θ))
So, the polar equation of the ellipse, with eccentricity 1/2 and directrix y = a, is r = (3a/2) / (1 - (1/2) * cos(θ)).
To determine whether to use sin or cosine in the polar equation, you need to consider the orientation of the ellipse. If the major axis of the ellipse is aligned with the x-axis, use cosine. If the major axis is aligned with the y-axis, use sine. In this case, since the major axis lies along the x-axis, you should use cosine in the polar equation.