can you check my work?
Find a polar equation of an ellipse with its focus at the pole an eccentricity of e=1/4 and directrix at y=4.
answer: √(x^2 + y^2) / (y - 4) = 1/4
4√(x^2 + y^2) = y - 4
4r = r sin θ - 4
(sin θ - 4) r = 4
r = 4/(sin θ - 4)
To check your work, let's start by understanding the definition of an ellipse in polar coordinates. An ellipse is defined as the locus of points such that the sum of the distances from two fixed points, called the foci, is constant.
In this case, the focus is at the pole (r = 0) and the eccentricity (e) is given as 1/4. The eccentricity of an ellipse is the ratio of the distance between the foci and the length of the major axis. As the eccentricity is less than 1, we know this is an ellipse.
The given directrix is at y = 4. The directrix of an ellipse is a line such that the ratio of the distance from any point on the ellipse to the directrix to the distance from that point to the focus is constant.
To find the equation of this ellipse, we can use the polar form of the equation of an ellipse:
r = (d1 + d2) / 2
where d1 is the distance from any point (r, θ) on the ellipse to the focus and d2 is the distance from that point to the directrix.
Since the focus is at (0, 0), the equation becomes:
r = (r + d2) / 2
Substituting the distance from a point on the ellipse to the directrix, we have:
r = (r + y - 4) / 2
Simplifying, we get:
2r = r + y - 4
Combining like terms, we have:
r = y - 4
This is the correct polar equation of the given ellipse with the focus at the pole and the directrix at y = 4.
Therefore, your answer √(x^2 + y^2) / (y - 4) = 1/4 is incorrect. The correct equation is r = y - 4.