can you check my work?
Find a polar equation of a hyperbola with its focus at the pole, an eccentricity of e=5/4 and directrix at y=2.
answer: r=ek/1+esintheta
r=(5/4*2)/1+5/4sin theta
r=5/2(1+(5sintheta)/4)
Hmmm. The standard form for the polar conics is
r = ep/(1 + e sinθ)
While your equation looks correct, I'd have written it as
r = (5/2) / (1 + (5/4) sinθ)
to make it clear what the parameters are. Or, if you dislike fractions,
r = 10/(4+5sinθ)
To check your work, let's go through the steps together:
1. Start with the given information: focus at the pole, eccentricity e = 5/4, and directrix at y = 2.
2. Recall the polar equation of a hyperbola in general form: r = e * (k / (1 + e * cos(theta))).
3. Since the focus is at the pole, k (the distance from the pole to the directrix) is equal to the distance from the pole to the focus, which is 2 units. Therefore, k = 2.
4. Plug in the given eccentricity and the value of k into the polar equation: r = (5/4) * (2 / (1 + (5/4) * cos(theta))).
It seems like you made a mistake in your answer. The correct equation should be:
r = (5/4) * (2 / (1 + (5/4) * cos(theta))).
Please double-check your steps to see if there were any errors in your calculations.