   Chapter 2.3, Problem 66E

Chapter
Section
Textbook Problem

The figure shows a fixed circle C1, with equation (x – 1)2 + y2 = 1 and a shrinking circle c, with radius r and center the origin. P is the point (0, r ), Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the x-axis. What happens to R as C2 shrinks, that is, as r →0+? To determine

To find: The point R when the circle C2 shrinks (as r0+).

Explanation

Given:

The equation of the fixed circle C1 is (x1)2+y2=1.

The equation of the shrinking circle C2, x2+y2=r2.

P be the point (0,r), Q be the upper point of intersection of the two circles and R be the point of intersection of the line PQ and the x axis.

Formula used:

Slope of the line: m=y2y1x2x1

Equation of line between two points: (yy1)=m(xx1)

Difference of square formula: (a2b2)=(a+b)(ab)

Limit Laws:

Suppose that c is a constant and the limits limxaf(x) and limxag(x) exist. Then,

Limit law 1: limxa[f(x)+g(x)]=limxaf(x)+limxag(x)

Limit law 2: limxa[f(x)g(x)]=limxaf(x)limxag(x)

Limit law 3: limxa[cf(x)]=climxaf(x)

Limit law 7: limxac=c

Limit law 9: limxaxn=an where n is a positive integer.

Limit law 11: limxaf(x)n=limxaf(x)n where n is a positive integer, if n is even, assume that limxaf(x)>0.

Calculation:

Given that the point Q is the point of intersection of the circles C1 and C2.

(x1)2+y2=1 (1)

x2+y2=r2 (2)

Eliminate y from equation (1) and (2),

(x1)2+y2(x2+y2)=1r2(x1)2x2=1r2x22x+1x2=1r22x+1=1r22x=r2

Isolate, x.

2x=r2x=r22

Substitute x=r22 in equation (2),

(r22)2+y2=r2r44+y2=r2y2=r2r44y2=r2(1r24)

Take square root on both sides,

y=r1r24

Therefore, the point Q is (12r2,r114r2).

Thus, the coordinates of P(x1,y1) and Q(x2,y2) are (0,r) and (12r2,r114r2).

Obtain the slope of the line passing through the points P and Q by using the slope of the line formula.

m=r114r2r12r20=r(114r21)12r2=2(114r21)r

Obtain the equation of the line joining the points P and Q by using the equation of line between two points as follows.

(yy1)=m(xx1)yr=2(114r21)r(x0)

yr=2(114r21)rx (3)

Note that, R is the point of intersection of the line PQ and the x axis

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