$\"\"$

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The equation is $\"\"$ .

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$\"\"$

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$\"\"$

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Differentiate 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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$\"\"$

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Find orthogonal trajectory :

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Orthogonal family has negative reciprocal slope.

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$\"\"$

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Apply cross multiplication.

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$\"\"$

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Integrate on each side.

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$\"\"$

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Integrate power rule: $\"\"$ .

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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 several values of $\"\"$ .

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Graph :

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$\"\"$

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Observe the graph:

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$\"\"$ are ellipses.

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$\"\"$ are curves.

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Graphs will vary.

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$\"\"$

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Graph :

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$\"\"$

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$\"\"$ are ellipses.

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$\"\"$ are curves.

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Graphs will vary.