   Chapter 2.6, Problem 52E

Chapter
Section
Textbook Problem

# 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. y = a x 2 , x 2 + 3 y 2 = b

To determine

To show:

Two families of curves are orthogonal trajectories of each other and sketch both families

Explanation

1) Given:

y=ax3, x2+3y2=b

2) Calculation:

Differentiating y=ax3, implicitly with respect to x we get,

dydx=3ax2

Let (x1,y1) be a point of intersection of both the curves

Therefore, at point of intersection slope becomes

dydx=3ax12

Now consider the curve  x2+3y2=b.

Now differentiating  x2+3y2=b implicitly with respect to x we get,

2x+6ydydx=0

6ydydx=-2x

dydx=-

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