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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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BuyFindarrow_forward

Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
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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