   Chapter 2, Problem 4P

Chapter
Section
Textbook Problem

The figure shows a point P on the parabola y = x2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.

To determine

To find: The position of Q when P approaches to origin and the limiting position of Q.

Explanation

Result used:

Slope of Perpendicular lines: The product of slope of the two perpendicular lines equal to –1.

Graph:

Calculation:

The perpendicular bisector of OP intersects the y-axis as P approaches the origin along the parabola then the length of the Q will increases and joins at a point R on the parabola.

Let R be the midpoint of OP, since P(x,x2) then R(x2,x22).

Let Q(0,a) and from figure 1that is OPQR.

The slope of OP is

mOP=x20x0=x2x=x

By the slopes of the perpendicular lines

mQR×mOP=1mQR=1mOP

mQR=1x (1)

Let Q(0,a)

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