A hyperbola as a constant difference of distances

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two vertices.

In the simple case of a horizontal hyperbola centred on the origin, we have the following:

1. $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

2. $c = \sqrt{a^2 + b^2} = a\varepsilon = a\sqrt{1 + \frac{b^2}{a^2}}$

The foci lie at $(\pm c, 0)$.

Now, if I'm not wrong about that, then this should be pretty basic algebra, but I can't see how to get from the above to an equation given a point $(x,y)$ describing the difference in distances to the foci as being $2a$. While I actually do care about the final result, how to get there is more important.

Why do I want to know this? Well, I'd like to attempt trilateration based off differences in distance rather than fixed radii.

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What is "an equation that results in $2a$ given x and y"? –  anon Jul 5 '11 at 1:29
@anon I've reworded that bit in a way that might make more sense. –  Iskar Jarak Jul 5 '11 at 2:13

We will use a little trick to avoid work. We want to have $$\sqrt{(x+c)^2+y^2} -\sqrt{(x-c)^2+y^2}=\pm 2a.\qquad\text{(Equation 1)}$$

Rationalize the numerator, by multiplying "top" and "bottom" by $\sqrt{(x+c)^2+y^2} +\sqrt{(x-c)^2+y^2}.$ After the (not very dense) smoke clears, we get $$\frac{4xc}{\sqrt{(x+c)^2+y^2} +\sqrt{(x-c)^2+y^2}}=\pm 2a.$$ Flip it over, do some easy algebra. We get $$\sqrt{(x+c)^2+y^2} +\sqrt{(x-c)^2+y^2}=\pm \frac{2cx}{a}.\qquad\text{(Equation 2)}$$ From Equations 1 and 2, by adding, we get $$2\sqrt{(x+c)^2+y^2}=\pm 2\left(a+ \frac{xc}{a}\right).$$ Cancel the $2$'s, square. We get $$x^2+2cx+c^2+y^2=a^2+ 2cx+ \frac{c^2x^2}{a^2}.$$ Now it's basically over, the $2cx$ terms cancel. Multiply through by $a^2$, put $c^2=a^2+b^2$, and rearrange.

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+1 much better than mine –  Ross Millikan Jul 5 '11 at 3:33
@Ross Millikan: Not really. But "rationalizing the numerator" does have a number of uses, so it is nice to be able to mention it. –  André Nicolas Jul 5 '11 at 3:38
Thanks, that was a big help. Totally forgot about that rationalising trick. –  Iskar Jarak Jul 5 '11 at 3:51

If we write the equation for what you said, a point $(x,y)$ on the hyperbola, taking $x \gt 0$ for convenience, must have $\sqrt{(x+c)^2+y^2}-\sqrt{(x-c)^2+y^2}=2a$. Squaring, $(x+c)^2+2y^2+(x-c)^2-2\sqrt{((x-c)^2+y^2)((x+c)^2+y^2)}=4a^2$. Then if you isolate the radical and square again, you should be able to cancel a lot of terms and get to the form you want.

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This was still helpful although not as detailed as the answer I accepted - I'd upvote if I could. –  Iskar Jarak Jul 5 '11 at 3:51

I personally like both of the above methods but there is a simpler way(in my opinion): If you use the two equations above( $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $c^2 =a^2+b^2$) and we know both $a$(since $2a$ is the difference in distances to the foci) and $c$ (where the foci lie) then just substitute $b^2$ with $c^2-a^2$ to get $\frac{x^2}{a^2} - \frac{y^2}{c^2-a^2} = 1$. I like it only because it's pretty succinct, you don't really repeat except for $a^2$ twice, you can use it even if the center of the hyperbola isn't at $(0,0)$ and there's basically no hard maths to explain.

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