How to satisfy this identity (conformal model in geometric algebra)

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SUMMARY

The discussion centers on solving the equation xy - √((x² + a²)(y² + c²)) = -1/a² - 1/c², where a² and c² are real constants. The proposed solution is x = (1 - a²)/2 and y = (1 - c²)/2. Participants explore whether a systematic method exists for deriving this solution or if trial and error is necessary. Suggestions include manipulating the equation by isolating xy and squaring both sides to simplify the problem.

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Hello,

I have the following equation in x and y: xy - \sqrt{(x^2+a^2)(y^2+c^2)} = -\frac{1}{a^2}-\frac{1}{c^2} where the quantities a2 and c2 are given real constants, and I have to find real values for x, and y such that the equation above is always satisfied.

Actually, I know that the solution should be: x = \frac{1-a^2}{2}, y = \frac{1-c^2}{2}, but I would like to know if there is a "mechanical" procedure to arrive at that solution, or if one has just to do "trial and error".

PS: for anyone interested, solving this equation is useful for constructing the conformal model in Geometric Algebra.
 
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Have you tried putting xy on the other side and then squaring both sides? The (xy)2 term will cancel.
 

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