Find the polar equation of an ellipse with a focus at the pole and major axis endpoints (4, 0)
and (2, 𝞹).
Thank you!
To find the polar equation of an ellipse with a focus at the pole and major axis endpoints, we first need to understand the standard polar equation for an ellipse.
The standard polar equation for an ellipse with a focus at the pole and major axis endpoints can be written as:
r = (d / (1 + e * cos(theta)))
Where:
- r is the distance from the pole to a point on the ellipse
- d is the distance from the focus to the directrix (which is equal to the distance from the focus to the pole)
- e is the eccentricity of the ellipse
- theta is the angle formed by the line connecting the pole to a point on the ellipse and the positive x-axis
In this case, the focus is at the pole and the major axis endpoints are (4, 0) and (2, π).
To find the distance from the focus to the directrix (d), we can use the distance formula:
d = sqrt((4 - 2)^2 + (0 - π)^2)
d = sqrt(2^2 + π^2)
d = sqrt(4 + π^2)
Next, we need to find the eccentricity (e) of the ellipse. The eccentricity can be calculated using the formula:
e = sqrt(1 - (b^2 / a^2))
where 'a' is the distance from the focus to a vertex (in this case the distance from the focus to (4,0)), and 'b' is the distance from the focus to the minor axis endpoints.
In this case, 'a' is equal to 4 and 'b' is equal to 2. Plugging these values into the eccentricity formula:
e = sqrt(1 - (2^2 / 4^2))
e = sqrt(1 - (4/16))
e = sqrt(1 - 1/4)
e = sqrt(3/4)
e = sqrt(3) / 2
Now we have all the necessary values to write the polar equation of the ellipse with a focus at the pole and major axis endpoints:
r = ((sqrt(4 + π^2)) / (1 + (sqrt(3) / 2) * cos(theta)))
Let me know if you need any further explanation!