Question

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)

Answer 1

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)