Chapter 10.1, Problem 94E

To determine

To determine whether the statement is true or false

Answer to Problem 94E

The given statement is True.

Explanation of Solution

Given information : 

The surface consisting of all points $\left(x,y,z\right)$ in space that are the same distance $r$ from the point $\left(h,k,l\right)$ has a circle as its $xy-trace$ .

Concept used:

If any surface has $xy-trace$ then every point lying in $xy-plane$ has a $z-coordinate$ zero.

Explanation: 

Equation of the surface consisting of all points $\left(x,y,z\right)$ in space that are the same distance $r$ from the point $\left(h,k,l\right)$ is

  ${\left(x-h\right)}^2+{\left(y-k\right)}^2+{\left(z-l\right)}^2=r^2$

To find $xy-trace$ put $z=0$

  $\begin{array}{l}{\left(x-h\right)}^2+{\left(y-k\right)}^2+{\left(0-l\right)}^2=r^2\\ {\left(x-h\right)}^2+{\left(y-k\right)}^2+{\left(l\right)}^2=r^2\\ {\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2-l^2=R^2\left(R^2=r^2-l^2\right)\end{array}$

The extracted equation resembles equation of circle with radius R.

Hence,

Given statement: The surface consisting of all points $\left(x,y,z\right)$ in space that are the same distance $r$ from the point $\left(h,k,l\right)$ has a circle as its $xy-trace$ is True.