   Chapter 11.6, Problem 44E

Chapter
Section
Textbook Problem

# Finding an Equation of a SurfaceIn Exercises 45 and 46, find an equation of the surface satisfying the conditions, and identify the surface.The set of all points equidistant from the point ( 0 , 0 , 4 ) and the xy-plane

To determine

To calculate: A surface contains the set of all point that are equidistant from the point (0,0,4) and the xy plane, find the equation of surface satisfying the provided conditions.

Explanation

Given:

The required surface contains the set of all point that are equidistant from the point (0,0,4) and the xy plane.

Formula used:

The distance between 2 points (x1,y1,z1) and (x2,y2,z2) is given by

D=(x2x1)2+(y2y1)2+(z2z1)2

Distance between a point and a plane is given by

D=|ax0+by0+cz0+d|a2+b2+c2

Calculation:

For the point (0,0,4) and the plane z=0.

Take a point be (x,y,z) which is equidistant mentioned point and planegiven plane is xy plane,

Hence its equation will be

z=0

Normal vector n=a,b,c=0,0,1

Let us now, equate the distances of the point (x,y,z) to the point (0,0,4) and the plane z=0 as

(x0)2+(y0)2+(z4)2=|0x+0y+1z+0|12x2+y2+(z4)2=±z2

As there is possibility of both the negative and coefficients for z variable at right hand side

Hence solve both cases

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