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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

Tangent lines T 1 and T 2 are drawn at two points P 1 and P 2 on the parabola y = x 2 and they intersect at a point P. Another tangent line T is drawn at a point between P 1 and P 2 ; it intersects T 1 at Q 1 and T 2 at Q 2 . Show that

| P Q 1 | | P P 1 | + | P Q 2 | | P P 2 | = 1

To determine

To show:

PQ1PP1+PQ2PP2=1

Explanation

1) Concept:

T1 and T2 are drawn at two points P1 and P2 on the parabola y=x2. Slope of tangent is the derivative of the given curve. Using this to find equation of both tangents.

By equating both tangents find point of intersection. By using distance formula find values of PQ1,  PQ2,  PP1 and |PP2|.

2) Formula:

(i) Slope of tangent =

mtanΑ=dydx

(ii) Equation of tangent through a, b is

y-b=mtanΑx-a

(iii) Distance between two points x1, y1 and (x2, y2) is given by

d=x2-x12+y2-y12

3) Given:

Tangent lines T1 and T2 drawn at points P1 and P2 on parabola y=x2, intersect at a point P. Another tangent line is drawn at a point between P1 and P2, which intersects T1 at Q1 and T2 at Q2.

4) Calculations:

Slope of tangent to the curve y=x2 is dydx=2x,

Therefore general equations of tangent to the curve at point Say P1a, a2 and P2b,b2, are

y-a2=2ax-a that is y=2ax-a2 and y=2bx-b2,

At point of intersection P, 2ax-a2=2bx-b2

Therefore, 2a-bx= a2-b2

Hence, x=a+b2,

Using this in any one equation, y =2aa+b2-a2=ab.

Therefore, point of intersection is,Pa+b2, ab.

Let Q  be point on the curve between P1 and P2 with co-ordinates, say c, fc. Let T be a tangent to the curve at Q

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