$\"\"$

\

The equation is $\"\"$ .

\

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

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

Find orthogonal trajectory.

\

Orthogonal family has negative reciprocal slope.

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

Apply integral on each side.

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

$\"\"$

\

Graph the two families.

\

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

\

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

\

Graph :

\

$\"\"$

\

Observe the graph:

\

$\"\"$ are hyperbola.

\

$\"\"$ are curves.

\

$\"\"$

\

Graph :

\

$\"\"$

\

Observe the graph:

\

$\"\"$ are hyperbola.

\

$\"\"$ are curves.