Find a polar equation of a hyperbola with its focus at the pole, an eccentricity of e=5/4 and directrix at y=2.
To find the polar equation of a hyperbola with its focus at the pole, eccentricity e, and directrix at a given line, we can use the general equation for a hyperbola in polar coordinates:
r = (ep)/(1 ± ecosθ)
where r is the distance from the focus, p is the distance from the pole to the directrix, e is the eccentricity, and θ is the angle between the polar axis and the line connecting the pole and a point on the hyperbola.
In this case, the focus is at the pole, which means the distance from the focus to the pole is zero. Therefore, p = 0.
The eccentricity e is given as 5/4.
The directrix is at y = 2, which is a horizontal line. In polar coordinates, the equation of a horizontal line is r = dsecθ, where d is the distance from the pole to the line, and secθ is the reciprocal of the cosine of θ.
Since the directrix is at y = 2, the distance d from the pole to the line is 2.
Now, substitute the values of p, e, and d into the general equation of the hyperbola in polar coordinates:
r = (e * p) / (1 ± ecosθ)
Since p = 0, the equation becomes:
r = 0 / (1 ± ecosθ)
Simplifying further, we have:
r = 0
Therefore, the polar equation of the hyperbola with its focus at the pole, eccentricity e = 5/4, and directrix at y = 2 is r = 0, which describes the pole itself.