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Determining a hyperbola from four points

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Determining a hyperbola from four points

I have these points to a hyperbola and I have it in terms of x(t) and y(t). It contains (0,0), (cos theta, sin theta), (c , d) and (e , f). Instead of two parametrically defined functions I would like it in standard form.

(x - h)^2 ��(y - k)^2
--------- - --------- = 1
a^2 ����������b^2

with the constants h, k, a, b in terms of c, d, e, f, and in trig functions of theta. Any thoughts?

Mitchell
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Determining a hyperbola from four points

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Determining a hyperbola from four points

I have two parametric equations for a hyperbola that I would like converted to standard form. If the parameter could be eliminated that would be a great help.

x(t)=sin^2(t)/cos(2t-b)

y(t)=sin(t)*cos(t)/cos(2t-b)

where for certain b is 60 degrees or pi / 3 radians. The general case where b is an unknown would also be a great help. If the conversion turns out to be impossible(as I suspect for the general case) would the equation of the line for the asymptotes be determinable instead?

Mitchell
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Determining a hyperbola from four points

Elimination of 't'-simply

This illustrates how to eliminate "t" without nearly the complexity anticipated.
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