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Find the orthogonal trajectories of the family of all circles passing through the points (1,0) and (-1,0)

Find the orthogonal trajectories of the family of curves consisting of all circles passing through the points (1,0) and (-1,0).


In a previous exercise (section 8.22, Exercise #11) we found that the family of all circles passing through the points (1,0) and (-1,0) satisfy the differential equation

    \[ (x^2 - y^2 -1) y' - 2xy = 0 \quad \implies \quad y' = \frac{2xy}{x^2-y^2-1}. \]

Therefore, the orthogonal trajectories satisfy the differential equation

    \[ y' = \frac{2xy}{y^2 - x^2 + 1}. \]

Incomplete.

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