   Chapter 10.6, Problem 38E 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 = 0 ,   0 ,   0 + n 1 ,   1 ,   2 intersects the sphere x 2 + y 2 + z 2 = 54 in two points. Find each point.

To determine

To explain:

Find the point of intersection of the line x, y, z=0, 0, 0+n1, 1, 2 and a sphere x2+y2+z2=54.

Explanation

The given line is x, y, z=0, 0, 0+n1, 1, 2.

Write the point form of the line.

x, y, z=0+n, 0+n, 0+2n

Simplifying,

x, y, z=n, n, 2n

From this,

x=n

y=n

z=2n

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

The equation of the sphere is x2+y2+z2=54.

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

n2+n2+2n2=54

Expanding the equation,

n2+n2+4n2=54

Simplifying,

6n2=54

Dividing by 6.

66n2=546

Cancelling the common factors,

n2=9

Finding the roots,

n=±3

Substitute n=3 in the point form of the line equation

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