$\"\"$

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The function is $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Find orthogonal trajectories:

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$\"\"$ represents the slope of the given family of curves at $\"\"$ .

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Orthogonal family has the negative reciprocal slope $\"\"$ .

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The function is $\"\"$ .

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$\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Apply integral on each side.

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Graph the two families.

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Graph the function $\"\"$ for several values of $\"\"$ .

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Graph the orthogonal function $\"\"$ for different values of $\"\"$ .

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Graph:

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$\"\"$

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The curves represent the given family of curves.

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The parabola represent the orthogonal trajectories.

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Graphs will vary.

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$\"\"$

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Graph:

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$\"\"$

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Lines: $\"\"$ .

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Parabolas: $\"\"$ .

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Graphs will vary.