Explanation Given information : Given parabola y = 1 − x 2 and the triangle A B C formed by the x − axis and the tangent line at P and Q is an equilateral triangle. From the given figure we can observe that the y = 1 − x 2 is symmetric about y − axis Which implies the tangents formed at the points P and Q on the parabola also symmetric about y − axis . Which implies, if Q = ( a , b ) be a point on the parabola then reflection of Q = ( a , b ) along y − axis is given by P = ( − a , b ) . Now we find equation of tangent passing through the point Q = ( a , b ) , Which is given by y − b = m ( x − a ) , where slope of the tangent, which is equal to derivative of parabola at Q = ( a , b ) ⇒ y | = − 2 x ⇒ m = − 2 a Thus, the required tangent equation A C is y − b = − 2 a ( x − a ) Where the coordinate of point C is when the tangent intersect at x − axis . ⇒ − b = 2 a ( x − a ) as y = 0 on x − axis ⇒ − b = 2 a x − 2 a 2 x = 2 a 2 − b 2 a Thus, the coordinate of point C is ( 2 a 2 − b 2 a , 0 ) . Similar way the coordinate of point A is when the tangent intersect at y − axis . ⇒ y − b = 2 a ( 0 − a ) as x = 0 on y − axis . ⇒ y − b = − 2 a 2 y = − 2 a 2 + b

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ISBN: 9781305177291

Author: James Stewart, Troy Day

Publisher: Cengage Learning

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Chapter 3, Problem 94E

To determine

The points P and Q on the parabola y=1−x2 so that the triangle ABC formed by the x−axis and the tangent line at P and Q is an equilateral triangle.

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Chapter 3, Problem 93E

Chapter 3, Problem 1CS

Chapter 3 Solutions

Biocalculus: Calculus For Life Sciences

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The points P and Q on the parabola y = 1 − x 2 so that the triangle A B C formed by the x − axis and the tangent line at P and Q is an equilateral triangle.

Solution Summary: The author explains that the triangle ABC formed by the x-axis and the tangent line at P and Q is an equilateral triangle.

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The points P and Q on the parabola y=1−x2 so that the triangle ABC formed by the x−axis and the tangent line at P and Q is an equilateral triangle.

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Chapter 3 Solutions