Find ( $c$ is an arbitrary constant ) orthogonal trajectories of circles:
$$ (x-c)^2+ y^2= 1 $$
2026-03-25 14:33:17.1774449197
Orthogonal trajectories of unit circles centered on x-axis
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$$\pm x+k=\int dy\sqrt{\frac1{y^2}-1}$$Put $y=\cos\theta$,$$\pm x+k=-\int \sin\theta\tan\theta~d\theta=-\int\frac{1-\cos^2\theta}{\cos\theta}d\theta\\=\sin\theta-\ln|\sec\theta+\tan\theta|\\=\sqrt{1-y^2}-\ln\left|1/y+\sqrt{1/y^2-1}\right|$$