   Chapter 4.7, Problem 55E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Let a and b be positive numbers. Find the length of the shortest line segment that is cut off by the first quadrant and passes through the point (a, b).

To determine

To find: The length of shortest line segment which is cut off by the first quadrant and passes through the point (a,b).

Explanation

Result used:

Let two points are (x1,y1) and (x2,y2)

The distance between two points =(x1x2)2+(y1y2)2

Calculation:

A line segment that passes through (a,b) with slope m where m<0

So, the equation of the line is

yb=m(xa)yb=mxma

Substitute x=0 in yb=mxma

yb=0×xmay=bma

The intercept of  y=bam

Substitute y=0 in yb=mxma

0b=mxmamx=mabx=abm

The intercept of x=abm

So,

A=(0,bam)B=(abm,0)

The length of the segment will be

d=(abm)2+(bam)2

Now, let S=d2, then

S=a22abm+b2m2+a2m22abm+b2

Differentiate with respect to m

dSdm=2abm22b2m3+2a2m2ab=2m3(amb)(b+am3)

For critical points, dSdm=0

2m3(amb)(b+am3)

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