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I have two large equations, both of the same form which I am trying to find the intersections of. The equations are: $$ f(x, y) = \frac{\frac{x ^ 2}{r_1} + \frac{y ^ 2}{r_2}} {1 + \sqrt{1 - \frac{p_1 x ^ 2}{ r_1 ^ 2} - \frac{p_2 y ^ 2}{r_2 ^ 2}}} + b \\ g(x, y) = \frac{\frac{x ^ 2}{s_1} + \frac{y ^ 2}{s_2}} {1 + \sqrt{1 - \frac{q_1 x ^ 2}{ s_1 ^ 2} - \frac{q_2 y ^ 2}{s_2 ^ 2}}} + c $$ What I seek is a two-dimensional equation $h(x)$ where for any value of $x$ the value of $y$ is provided. I have proceeded by attempting to solve $$ f(x, y) - g(x, y) = 0 $$ for $y$. For example in a much easier case, if $$ f(x, y) = ax ^ 2 + by ^ 2 + c \\ g(x, y) = dx ^ 2 + ey^2 + f $$ then $$h(x) = \sqrt{\frac{f-c-x^2(a-d)}{b-e}}$$ I have tried to do this with pen and paper then with Wolfram Alpha then with SymPy but I can't get anything! Any help would be greatly appreciated. I don't even know if it is possible or not so any advice or references would be warmly received.

Asaf Karagila
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Danyule
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    You could look at the implicit function theorem. – Nameless Dec 13 '13 at 14:13
  • As a matter of terminology, we don't say two functions intersect. Their graphs or solution sets can intersect, or you can ask where the functions agree. – Kimball Jan 24 '15 at 11:53

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