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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Suppose that three points on the parabola y = x2 have the property that their nom1al lines intersect at a common point. Show that the sum of their x-coordinates is 0.

To determine

To show: The sum of their x coordinate is zero.

Explanation

Given:

The parabola equation is, y=x2.

Calculation:

Given that three points on the parabola have the property that their normal line intersect at common point.

Consider the three point (a1,a12), (a2,a22) and (a3,a32).

If one of the ai is zero (suppose a1=0), by symmetry a2=a3, then a1+a2+a3=0.

Suppose that any of ai0 where i=1,2,3,

The slope of the tangent to the equation y=x2 is, dydx=2x.

That is, the slope of the tangent to the equation y=x2 at (ai,ai2) is, dydx=2ai.

Thus, the slope of the normal line to the equation y=x2 at (ai,ai2) is, m=12ai

The equation of the normal line at (ai,ai2) is computed as follows,

yai2=12ai(xi)y=ai2+12ai(xai)

Obtain the x coordinate of the intersection of the normal line from (a1,a12) and (a2,a22).

Since y=a12+12a1(xa1) and y=a22+12a2(xa2) implies that

a12+12a1(xa1)=a22+12a2(xa2)a1212a1x+12=a2212a2x+12x2a2x2a

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