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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.5, Problem 42E

To determine

**To show:** One family of the curves are orthogonal trajectories to the other family.

Expert Solution

**Given:**

The equation of the curves

**Derivative rules:**

*Chain rule*:

**Proof:**

The two family of the circles

Clearly, the tangent line to the family of curves *a* and *b* are nonzero.

Therefore, the two family of the circles are orthogonal trajectories of each other at

Consider two circle is intersect at non zero point

Substitute

Differentiate *x*,

Apply the chain rule and simplify the terms,

Thus, the derivative of the equation

That is, the slope of the tangent to

Differentiate *x*,

Apply the chain rule and simplify the terms,

Thus, the derivative of the equation

That is, the slope of the tangent to

**Note:** The two tangent lines are orthogonal if the product of their slopes is -1.

Substitute the equation (1) and (2),

Therefore, the family of the curves

Hence the required result is proved.

**Graph:**

The sketch of the family of the circles

From the graph, it is observed that the two family of the circles