   Chapter 12, Problem 38RE

Chapter
Section
Textbook Problem

# A surface consists of all points P such that the distance from P to the plane y = 1 is twice the distance from P to the point (0, −1, 0). Find an equation for this surface and identify it.

To determine

To find and identify: An equation of the surface.

Explanation

Given data:

Distance P to the point (0,1,0) and plane is y=1 .

Formula used:

Consider the standard equation of an ellipsoid.

x2a2+y2b2+z2c2=1 (1)

Consider point P(x,y,z) .

The distance from (D) P(x,y,z) to (0,1,0) .

D=(x0)2+(y+1)2+(z0)2D=(x)2+(y+1)2+(z)2

As the plane y=1 is twice the distance from P(x,y,z) to the point (0,1,0) that is,

|y1|=2D

Substitute (x)2+(y+1)2+(z)2 for D ,

|y1|=2((x)2+(y+1)2+(z)2)(y1)2=4[(x)2+(y+1)2+(z)2](y1)2=4(x)2+4(y+1)2+4(z)2y22y+1=4x2+4(y2+2y+1)+4z2

Rearrange the equation

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