   Chapter 10.6, Problem 37E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

The line x ,   y ,   z = 3 ,   4 ,   5 + n 3 ,   4 ,   - 5 intersects the sphere x 2 + y 2 + z 2 = 100 in two points. Find each point.

To determine

To explain:

Find the point of intersection of the line x, y, z=3, 4, 5+n3, 4, -5 and a sphere x2+y2+z2=100.

Explanation

The given line is x, y, z=3, 4, 5+n3, 4, -5.

Write the point form of the line.

x, y, z=3+3n, 4+4n, 5-5n

From this,

x=3+3n

y=4+4n

z=5-5n

It is given that the line intersects the sphere in two points.

The equation of the sphere is x2+y2+z2=100.

Substitute the value of x, y, z in the sphere equation.

3+3n2+4+4n2+5-5n2=100

Expanding the equation,

32+3n2+2·3·3n+42+4n2+2·4·4n+52+5n2-2·5·5n

Multiplying,

9+9n2+18n+16+16n2+32n+25+25n2-50n=0

Combining the like terms,

50n2+50-100=0

Simplifying,

50n2-50=0

Taking 50 as common,

50n2-1=0

n2-1=0

n2=1

Finding the roots,

n=±1

Substitute n=1 in the point form of the line equation

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