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    A hyperbola of the form
    x^2/(alpha)^2 - y^2/β^2 =1
    has asymptotes with equation y^2=m^2x^2 and passes through the point (a,0). Find an euqation of the hyperbola in terms of x,y,a and m.

    A point P on this hyperbola is equidistant from one of its asymptotes and the x-axis. Prove that, for all values of m, P lies on the curve with equation
    (x^2-y^2)^2=4x^2(x^2-a^2)
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    (Original post by totaljj)
    A hyperbola of the form
    x^2/(alpha)^2 - y^2/β^2 =1
    has asymptotes with equation y^2=m^2x^2 and passes through the point (a,0). Find an euqation of the hyperbola in terms of x,y,a and m.

    A point P on this hyperbola is equidistant from one of its asymptotes and the x-axis. Prove that, for all values of m, P lies on the curve with equation
    (x^2-y^2)^2=4x^2(x^2-a^2)
    For the first bit I get x²/a² - y²/a²m² = 1

    Not too sure about the second bit... thought about doing it in parameters, so considering a point (a sect, m/a tant) then using the formula for the closest distance of a point to a line (which may be in the formula book) - the line being y = mx ( or maybe y = -mx it shouldn't matter) and equating it to m/a tant.
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    Okay for the second bit first write the equation in terms of parameters, that is x = a sect and y = ma tant

    Now the distance to the x-axis is just the y coordinate ie ma tant. The perpendicular distance of a point to a line is in the formula book so the distance of (a sect, ma tant) to mx - y = 0 is...

    (ma sect - matant)/sqrt(1 + m²) and from the question this is equal to ma tant. Squaring gives...

    (sect - tant)² = (1 + m²)tan²t

    ie sec²t - 2sect.tant = m²tan²t

    putting back in (x,y) gives x²/a² - 2xy/ma² = y²/a²

    Now the question asks for the equation for all m so we must find an expression independant of m. To do this rearrange the above to get m in terms of x, y and a...

    2xy/m = x² - y² so 1/m = (x² - y²)/2xy

    But the point lies on the hyperbola, so x² - y²/m² = a²

    sub in 1/m² gives x² - y²((x² - y²)/2xy)² = a²

    so 4x²y²x² - y²(x² - y²)² = 4a²x²y² cancelling y² and factorising gives...

    4x²(x² - a²) = (x² - y²)²

    Hope that helps (even if it is a bit late!)
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    thank you phill
 
 
 
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Updated: June 17, 2004
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