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The equation is $\"\"$ .

Differentiate on each side with respect to $\"\"$ .

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$\"\"$ .

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Find orthogonal trajectory :

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Apply cross multiplication.

$\"\"$ .

Integrate on each side.

$\"\"$ .

Integrate power rule: $\"\"$ .

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The function is $\"\"$ .

(1) Draw the coordinate plane.

(2) Graph the function $\"\"$ for several values of $\"\"$ .

Graph :

$\"\"$ .

The curves represent the given family of curves.

The ellicpse represent the orthogonal trajectories.

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Graph :

$\"\"$ .