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Math Help - prove that tangents meet on the diameter

  1. #1
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    Talking prove that tangents meet on the diameter

    This question came from my Calculus hw.

    The circle  x^2 + y^2 + 2gx + 2fy + k = 0 intersects the rectangular hyperbola  x = ct  y= \frac{c}{t} in four points at P_1 (ct_1, \frac{c}{t_1}), P_2(ct_2, \frac{c}{t_2}) , P_3(ct_3, \frac{c}{t_3}) , P_4(ct_4, \frac{c}{t_4}) .

    By showing that  t_1t_2t_3t_4 = 1, Prove that the tangents at P_1 and P_2 meet on the diameter of the hyperbola perpendicular to  P_3P_4

    I can't show that  t_1t_2t_3t_4 = 1. I think it has something to do with the angle in a semi-circle being 90 degrees or something. I can't do the next part either.

    Thank you in advance
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  2. #2
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    Quote Originally Posted by differentiate View Post
    This question came from my Calculus hw.

    The circle  x^2 + y^2 + 2gx + 2fy + k = 0 intersects the rectangular hyperbola  x = ct  y= \frac{c}{t} in four points at P_1 (ct_1, \frac{c}{t_1}), P_2(ct_2, \frac{c}{t_2}) , P_3(ct_3, \frac{c}{t_3}) , P_4(ct_4, \frac{c}{t_4}) .

    By showing that  t_1t_2t_3t_4 = 1, Prove that the tangents at P_1 and P_2 meet on the diameter of the hyperbola perpendicular to  P_3P_4

    I can't show that  t_1t_2t_3t_4 = 1. I think it has something to do with the angle in a semi-circle being 90 degrees or something. I can't do the next part either.
    For the first part, substitute  x = ct and  y= c/t in the equation of the circle. This will give a quartic equation in t, in which the product of the four roots is the constant term.
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