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Given:
e=d2/d1,
a^2+b^2=c^2,
(x/a)^2-(y/b)^2=1,
a graph of a hyperbola with its center on the origin,
a=distance b/w the center and the vertex of the hyperbola,
c=distance b/w the center and the focus,
d=distance b/w the center and the directrix,
d1=distance b/w the directrix and the vertex,
d2=distance b/w the vertex and the focus......

Prove:
e=a/d,
e=c/a,
d=a^2/c,
LR(Latus Rectum)=2b^2/a......

I hope the givens were clear enough, I did my best...
I suggest that you graph or sketch the hyperbola first.
I've proved LR myself, but the rest was just too hard(worked for literally 3 hrs and 30 min after dinner).

Thank you and please prove it clearly(if possible) so I could understand.

2007-04-12 09:08:09 · 1 answers · asked by Korean Pride 2 in Science & Mathematics Mathematics

1 answers

Let's start with the definition of eccentricity. It is the ratio of the distance from the focus to a point on the hyperbola, divided by the distance from the directrix to a point of the hyperbola. Since this ratio is the same for any point on either of the hyperbolas, let's look at the two vertices and apply the definition to get two equations.

e = (c - a) / (a - d) for the near vertex
e = (c + a) / (a + d) for the far vertex

Now set these two equations for e equal to each other.

e = (c - a) / (a - d) = (c + a) / (a + d)

Multiply thru by (a - d)(a + d) to clear the denominators.

(c - a)(a + d) = (c + a)(a - d)
ac + cd - a² - ad = ac - cd + a² - ad
cd - a² = - cd + a²

2cd = 2a²
cd = a²

Divide both sides by ad.

c/a = a/d

So far we have proved that if the two definitions given above for e are true, then the equation immediately above holds. But we have not as yet shown that either ratio equals e.

2e = (c - a) / (a - d) + (c + a) / (a + d)
2e = [(c - a)(a + d) + (c + a)(a - d)] / (a - d)(a + d)

2e = [ac + cd - a² - ad + ac - cd + a² - ad] / (a² - d²)
2e = 2(ac - ad) / (a² - d²) = 2a(c - d) / (a² - d²)
e = a(c - d) / (a² - d²)

But we know that

c/a = a/d
c = a²/d

Plug in for c.

e = a(c - d) / (a² - d²)
e = a(a²/d - d) / (a² - d²) = a(a² - d²) / [d(a² - d²)] = a/d

And we know

c/a = a/d

So

e = c/a = a/d

If there was an easier way I didn't find it.
_____________________

Solve for d.

c/a = a/d
cd = a²
d = a²/c
_____________

You said you already solved for LR.

2007-04-12 17:46:08 · answer #1 · answered by Northstar 7 · 0 0

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