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Math Help - Hyperbola Problem

  1. #1
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    Hyperbola Problem

    I'm having trouble with this problem. Any help would be appreciated.

    The normal at any point on the hyperbola x^2/a^2 - y^2/b^2 = 1 intersects the x and y axes at P and Q respectively. Find the locus of the midpoint of PQ.

    Thanks
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  2. #2
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    Quote Originally Posted by jeta View Post
    I'm having trouble with this problem. Any help would be appreciated.

    The normal at any point on the hyperbola x^2/a^2 - y^2/b^2 = 1 intersects the x and y axes at P and Q respectively. Find the locus of the midpoint of PQ.

    Thanks
    The hyperbola can be defined by the following parametric equations:

    x = a \sec t = \frac{a}{\cos t}

    y = b \tan t = \frac{b \sin t}{\cos t}.


    Then \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} = -\frac{b}{a} \frac{1}{\sin t}.


    The equation of the tangent can be written using the model y - y_1 = m (x - x_1): y - \frac{b \sin t}{\cos t} = -\frac{b}{a} \frac{1}{\sin t} \left( x - \frac{a}{\cos t} \right) .

    Use this equation to get the coordinates of the x- and y-intercepts, that is, the coordinates of P and Q. Use these coordinates to get the midpoint of PQ.


    You now have the parametric equations of the required locus. Convert them into a Cartesian equation if required.


    Warning: My carelessness is a well documented fact. The above results may contain errors. You should work through the calculations very carefully for yourself.
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  3. #3
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    Thanks alot for your help.
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