Polar equations of conics question

[demonelite123]

for this problem, I've been given the vertices of the hyperbola as (4, pi/2) and (-1, 3pi/2). the question asks to find the polar equation of this hyperbola.

so what i did was do a quick sketch of the graph. (4, pi/2) is essentially (0,4) and (-1, 3pi/2) is essentially (0,1). the midpoint of the two vertices is (0,2.5) so that's the center. since the focus is at (0,0), c = 2.5 and a = 1.5

therefore e = c/a and e = 5/3

so i wrote out the equation r = (5/3)p / 1 + (5/3)sin& since the directrix is above the x-axis or polar axis between the two vertices. (by the way I'm using "&" to stand for the greek letter theta.)

then i just used one of the points they gave me (4, pi/2) and plugged it into the equation to solve for p. 4 = (5/3)p / (1 + (5/3)) when i solve for p, i get that p = 32/5.

however when i plug in the other point they gave me for the hyperbola (-1, 3pi/2), i get a different p value. -1 = (5/3)p / (1 - (5/3)) and when i solve for p i get that p = 2/5.

what's going on? how come the p values are different? both points are vertices of the hyperbola so i don't know why I'm getting different answers. the back of my textbook says that p = 8/5. how did the book get that? and how am i getting two different p values? please help me! I'm so confused.

 

[HallsofIvy]

What are you using for the eccentricity? Assuming the vertices are also given in polar coordinates, the vertices are at (0, 4) and (0, -1) so the center is at (0,3/2) but there are an infinite number of hyperboles that fit that.

 

[demonelite123]

the vertices they have given me are (4, pi/2) and (-1, 3pi/2) which in rectangular coordinates is (0,4) and (0,1) so the center of the hyperbola is (0,5/2) and (0,0) is one focus of the hyperbola.

so using that info i determined that c = 2.5 (the distance from center to focus) and a = 1.5 (distance from center to vertex). and i know that e = c/a, so i did e = 2.5 / 1.5 = 5/3

so my eccentricity of this hyperbola is 5/3.

 

[gabbagabbahey]

so i wrote out the equation r = (5/3)p / 1 + (5/3)sin& since the directrix is above the x-axis or polar axis between the two vertices. (by the way I'm using "&" to stand for the greek letter theta.)

Hmmm... shouldn't this be:

$$r=\frac{\left(\frac{5}{3}\right)p}{-1+\left(\frac{5}{3}\right)\sin(\theta)}$$


?

 

[demonelite123]

hm in my textbook, they only show the formulas r = ep / (1 +/- cos(&)) and r = ep / (1 +/- sin(&))

(& stands for "theta")

how did you get the -1 in your equation?