Question

Two curves are orthogonal if their tangent lines are

perpendicular at each point of intersection. Show that the given

families of curves are orthogonal trajectories of each other; that

is, every curve in one family is orthogonal to every curve in the

other family. Sketch both families of curves on the same axes.

Answer 1

Step 1:

The curve equations are and .

Consider .

Apply derivative on each side with respect to .

Step 2:

Consider .

Now,

Apply derivative on each side with respect to .

Substitute in the above expression.

.

Now we can observe that product slope of and is equal to .

Therefore both the curves are orthogonal to each side.

Step 3:

Graph both the curves.

Consider different values of , and .

Graph each curve for different values of , and .

Solution :

The two curves and are orthogonal to each other.