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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Show that the length of the portion of any tangent line to the asteroid x 2 / 3 + y 2 / 3 = a 2 / 3 cut of by the coordinate axes is constant.

To determine

To show:

The length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant

Explanation

1.Formula:

(i) Chain rule:

dydx= dydududx

(ii) Power Rule:

ddxxn=n xn-1

(iii) Standard Formula:

ddxconstant=0

(iv) Slope-point form:

y-y1=m x-x1

(v) Use distributive property:

a (b+ c) = a*b + a*c

2. Given:

x2/3+y2/3=a2/3

3.Calculation:

Differentiating implicitly y with respect to x,

By using power rule

23x23-1+23y23-1dydx=0

23x-13+23y-13dydx=0

Subtracting

23x-13

from both sides

23y-13dydx=-23x-13

Cancel out the common terms

y-13dydx=-x-13

dydx=-x-13y-13

dydx=-yx1/3

dydx=slope=m=chnage in ychnage in x

m=-yx1/3

Equation of tangent line passing through point (x0,y0) on curve is,

y-y0=-y0x01/3x-x0

The tangent line intercepts the x axis at y=0 and at y axis at   x=0

Now solving for x intercept, substitute y=0

0-y0=-y0x01/3x-x0

-y0=-y0x01/3x-x0

Solving for x,

y0=y0x01/3x-x0

By using distributive property

y0=y0x01/3x-y0x01/3x0

y0x01/3x=y0+y0x01/3x0

Dividing the whole equation by y0x01/3

x=y0+y0x01/3x0y0x013

x=y0x0y01/3+x0

x=y01-13x01/3+x0

x=y023x01/3+x0

Now, (x0,y0) satisfies asteroid equation

x02/3+y0

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