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Given polar coordinates and the endpoints of its transverse axis.

$(3,0)$ & $(-15,\pi)$

Ok so as I understand it the first point is representing the vertex of one side of the hyperbola at $x=3$, $y=0$

My main issue is I can seem to figure out how to find the directrix of the equation. I'm using the knowledge that PF/PD = eccentricity $(e)$, which is sometimes written as $c/a = e$

I get that the second polar part is $r$ and $\theta$, but again finding the directrix is the problem here.

dantopa
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Giovanni
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  • You know where a focus and the end points of the axis are, so you should be able to compute $a$ and $c$, and so also the eccentricity. – amd May 19 '17 at 06:22

1 Answers1

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We’re given that the vertices of the hyperbola are at $3\angle0$ and $-15\angle\pi=15\angle0$, i.e., at Cartesian coordinates $(3,0)$ and $(15,0)$.

polar hyperbola

This means that $a=6$ (half of the distance between the vertices), the center of the hyperbola is at $(9,0)$ (the midpoint of the axis) and $c=9$. Each directrix is at a distance of $\frac{a^2}c$ from the center, which makes the one nearer the origin the line $x=9-\frac{36}9=5$. The distance of a point $r\angle\theta$ from this line is $5-r\cos\theta$ and the distance from the focus is simply $r$. I trust that you can take it from here.

amd
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    ah ok I was confused about the 2nd vertex being (-15, pi) but I now see that would be at pi radians and then flipped over the y axis because of the negative number and a magnitude of 15. – Giovanni May 19 '17 at 14:25