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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find points P and Q on the parabola y = 1 – x2 so that the triangle ABC formed by the x-axis and the tangent lines at P and Q is an equilateral triangle. (See the figure.)

images

To determine

To find: The points P and Q on the parabola.

Explanation

Given:

The equation of the parabola is, y=1x2.

The triangle ABC formed by the x-axis and tangent line at P and Q is an equilateral triangle.

Calculation:

Consider the point P is located at (xP,yP) and the point Q is located at (xQ,yQ).

Here, the triangle is equilateral which implies that the symmetry about y axis.

That is, xP=xQ and yP=yQ.

The derivative of the function y=1x2 is computed as follows,

dydx=ddx(1x2)=02x=2x

That is, the slope of the tangent to curve is, dydx=2x.

Since the triangle is equilateral, every angle is 60°

tan60°=oppositeOCAOOC=3

The slope of the tangent at Q is computed as follows,

m=tan120°=3

Since the slope of the tangent at Q(xQ,yQ) is, m=

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