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Find the orthogonal trajectories of the family of all circles passing through (1,1) and (-1,-1)

Find the orthogonal trajectories of the family of curves consisting of all circles passing through the points (1,1) and (-1,-1).


In a previous exercise (section 8.22, Exercise #12) we found that the family of all circles passing through the points (1,0) and (-1,0) satisfy the differential equation

    \[ (x^2 + 2xy - y^2 - 2)y' - y^2 - 2xy + x^2 + 2 = 0 \quad \implies \quad y' = \frac{y^2 + 2xy - x^2 - 2}{x^2 + 2xy - y^2 - 2}. \]

Therefore, the orthogonal trajectories satisfy the differential equation

    \[ y' = \frac{y^2 - x^2 - 2xy + 2}{y^2 - x^2 + 2xy - 2}. \]

Incomplete.

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