Modern Analytic Geometry

William Wooton, Edwin F. Beckenbach, Frank J. Fleming

Chapter 10

Surfaces and Transformations of Coordinates in Space - all with Video Answers

Educators

Section 1

Spheres

Problem 1

In Exercises 1-8, Find an equation of the sphere with the given radius $r$ and the given center $\mathbf{C}$.
$\mathbf{C}(5,0,-7) ; r=2$

Steven Clarke

Steven Clarke

Numerade Educator

Problem 2

Find an equation of the sphere with the given radius $r$ and the given center $\mathbf{C}$.
$\mathbf{C}(-2,0,3) ; r=4$

Linh Vu

Numerade Educator

Problem 3

Find an equation of the sphere with the given radius $r$ and the given center $\mathbf{C}$.
$\mathbf{C}(0,0,-1) ; r=5$

Thane Stiles

Numerade Educator

Problem 4

Find an equation of the sphere with the given radius $r$ and the given center $\mathbf{C}$.
$\mathbf{C}(-2,0,0) ; r=3$

Linh Vu

Numerade Educator

Problem 5

Find an equation of the sphere with the given radius $r$ and the given center $\mathbf{C}$.
$\mathbf{C}(-1,-2,2) ; r=7$

Thane Stiles

Numerade Educator

Problem 6

Find an equation of the sphere with the given radius $r$ and the given center $\mathbf{C}$.
$\mathbf{C}(3,-4,-3) ; r=6$

Steven Clarke

Numerade Educator

Problem 7

Find an equation of the sphere with the given radius $r$ and the given center $\mathbf{C}$.
$\mathbf{C}\left(\frac{3}{2}, \frac{2}{3},-\frac{1}{2}\right) ; r=\frac{1}{4}$

Steven Clarke

Numerade Educator

Problem 8

Find an equation of the sphere with the given radius $r$ and the given center $\mathbf{C}$.
$\mathbf{C}\left(\frac{1}{4},-\frac{1}{4}, \frac{3}{4}\right) ; r=\frac{1}{5}$

Steven Clarke

Numerade Educator

Problem 9

In Exercises 9-16, Identify the graph of the given equation.
$x^2+y^2+z^2-2 x+4 y+6 z+5=0$

Anthony Ramos

Numerade Educator

Problem 10

Identify the graph of the given equation.
$x^2+y^2+z^2+8 x-4 y-4 z+8=0$

Anthony Ramos

Numerade Educator

Problem 11

Identify the graph of the given equation.
$x^2+y^2+z^2+x+3 y=0$

Anthony Ramos

Numerade Educator

Problem 12

Identify the graph of the given equation.
$x^2+y^2+z^2-2 x+z=0$

Anthony Ramos

Numerade Educator

Problem 13

Identify the graph of the given equation.
$x^2+y^2+z^2+8 x-6 y+4 z+29=0$

Anthony Ramos

Numerade Educator

Problem 14

Identify the graph of the given equation.
$x^2+y^2+z^2+2 x-2 y-4 z+6=0$

Anthony Ramos

Numerade Educator

Problem 15

Identify the graph of the given equation.
$x^2+y^2+z^2+2 x-2 y-4 z+7=0$

Anthony Ramos

Numerade Educator

Problem 16

Identify the graph of the given equation.
$x^2+y^2+z^2+4 x+3 y-2 z+7=0$

Linh Vu

Numerade Educator

Problem 17

In Exercises 17-24, Determine an equation, the coordinates of the center, and the radius, of the sphere through the points $\mathbf{Q}, \mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ whose coordinates are given.
$\mathbf{Q}(0,0,0), \mathbf{R}(1,0,0), \mathbf{S}(0,1,0), \mathbf{T}(0,0,1)$

Eduard Sanchez

Numerade Educator

Problem 18

Determine an equation, the coordinates of the center, and the radius, of the sphere through the points $\mathbf{Q}, \mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ whose coordinates are given.
$\mathbf{Q}(0,0,0), \mathbf{R}(1,1,0), \mathbf{S}(1,0,1), \mathbf{T}(0,1,1)$

Eduard Sanchez

Numerade Educator

Problem 19

Determine an equation, the coordinates of the center, and the radius, of the sphere through the points $\mathbf{Q}, \mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ whose coordinates are given.
$\mathbf{Q}(1,1,1), \mathbf{R}(1,1,-1), \mathbf{S}(1,-1,1), \mathbf{T}(-1,1,1)$

Eduard Sanchez

Numerade Educator

Problem 20

Determine an equation, the coordinates of the center, and the radius, of the sphere through the points $\mathbf{Q}, \mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ whose coordinates are given.
$\mathbf{Q}(1,1,1), \mathbf{R}(1,-1,-1), \mathbf{S}(-1,1,-1), \mathbf{T}(-1,-1,1)$

Eduard Sanchez

Numerade Educator

Problem 21

Determine an equation, the coordinates of the center, and the radius, of the sphere through the points $\mathbf{Q}, \mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ whose coordinates are given.
$\mathbf{Q}(2,0,5), \mathbf{R}(0,0,1), \mathbf{S}(3,-1,5), \mathbf{T}(-1,-4,2)$

Eduard Sanchez

Numerade Educator

Problem 22

Determine an equation, the coordinates of the center, and the radius, of the sphere through the points $\mathbf{Q}, \mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ whose coordinates are given.
$\mathbf{Q}(-5,-4,1), \mathbf{R}(3,4,-5), \mathbf{S}(0,0,4), \mathbf{T}(0,8,0)$

Eduard Sanchez

Numerade Educator

Problem 23

Determine an equation, the coordinates of the center, and the radius, of the sphere through the points $\mathbf{Q}, \mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ whose coordinates are given.
$\mathbf{Q}(2,1,-3), \mathbf{R}(1,2,0), \mathbf{S}(0,2,-2), \mathbf{T}(0,0,3)$

Eduard Sanchez

Numerade Educator

Problem 24

Determine an equation, the coordinates of the center, and the radius, of the sphere through the points $\mathbf{Q}, \mathbf{R}, \mathbf{S}$, and $\mathbf{T}$ whose coordinates are given.
$\mathbf{Q}(5,-3,7), \mathbf{R}(2,0,1), \mathbf{S}(0,-2,0), \mathbf{T}(-2,1,2)$

Eduard Sanchez

Numerade Educator

Problem 25

In Exercises 25-32, Find an equation of the plane $\rho$ that is tangent at the given point T to the sphere $\delta$ whose equation is given.
$x^2+y^2+z^2=49 ; \mathbf{T}(6,2,-3)$

Linh Vu

Numerade Educator

Problem 26

Find an equation of the plane $\rho$ that is tangent at the given point T to the sphere $\delta$ whose equation is given.
$x^2+y^2+z^2=169 ; \mathbf{T}(-12,3,4)$

Linh Vu

Numerade Educator

Problem 27

Find an equation of the plane $\rho$ that is tangent at the given point T to the sphere $\delta$ whose equation is given.
$(x-4)^2+(y+2)^2+z^2=121 ; \mathbf{T}(-5,4,2)$

Steven Clarke

Numerade Educator

Problem 28

Find an equation of the plane $\rho$ that is tangent at the given point T to the sphere $\delta$ whose equation is given.
$x^2+(y-1)^2+(z+3)^2=25 ; \mathbf{T}(3,1,-7)$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 29

Find an equation of the plane $\rho$ that is tangent at the given point T to the sphere $\delta$ whose equation is given.
$(x+2)^2+(y+3)^2+(z-1)^2=10 ; \mathbf{T}(-3,-3,4)$

Raj Bala

Numerade Educator

Problem 30

Find an equation of the plane $\rho$ that is tangent at the given point T to the sphere $\delta$ whose equation is given.
$(x-1)^2+(y+1)^2+(z+3)^2=11 ; \mathbf{T}(0,0,0)$

Linh Vu

Numerade Educator

Problem 31

Find an equation of the plane $\rho$ that is tangent at the given point T to the sphere $\delta$ whose equation is given.
$x^2+y^2+z^2+2 x-4 y-2 z-15=0 ; \mathbf{T}(3,0,2)$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 32

Find an equation of the plane $\rho$ that is tangent at the given point T to the sphere $\delta$ whose equation is given.
$x^2+y^2+z^2-4 x+6 y+4 z+3=0 ; \mathbf{T}(0,-2,1)$

Linh Vu

Numerade Educator

Problem 33

Show that the tangent plane $\odot$ to the sphere $S$ with equation

$$
x^2+y^2+z^2+G x+H y+I z+J=0
$$

at the point $\mathbf{T}\left(x_1, y_1, z_1\right)$ on $\delta$ has equation

$$
x_1 x+y_1 y+z_1 z+\frac{G}{2}\left(x+x_1\right)+\frac{H}{2}\left(y+y_1\right)+\frac{I}{2}\left(z+z_1\right)+J=0 .
$$

Tanishq Gupta

Numerade Educator

Problem 34

Show that the sphere through the noncoplanar points $\mathbf{Q}\left(x_1, y_1, z_1\right)$, $\mathbf{R}\left(x_2, y_2, z_2\right), \mathbf{S}\left(x_3, y_3, z_3\right)$, and $\mathbf{T}\left(x_4, y_4, z_4\right)$ has equation

$$
\left|\begin{array}{lllll}
x^2+y^2+z^2 & x & y & z & 1 \\
x_1^2+y_1^2+z_1^2 & x_1 & y_1 & z_1 & 1 \\
x_2^2+y_2^2+z_2^2 & x_2 & y_2 & z_2 & 1 \\
x_3^2+y_3^2+z_3^2 & x_3 & y_3 & z_3 & 1 \\
x_4^2+y_4^2+z_4^2 & x_4 & y_4 & z_4 & 1
\end{array}\right|=0
$$

Nick Johnson

Numerade Educator

Problem 35

In Exercises 35 and 36, Identify the graph of the equation obtained by substituting the coordinates of the given points in the equation of Exercise 34. If the graph is a sphere, find its radius and the coordinates of its center.
$\mathbf{Q}(0,0,0), \mathbf{R}(1,0,0), \mathbf{S}(2,1,3), \mathbf{T}(5,2,6)$

Christina Shute

Christina Shute

Numerade Educator

Problem 36

Identify the graph of the equation obtained by substituting the coordinates of the given points in the equation of Exercise 34. If the graph is a sphere, find its radius and the coordinates of its center.
$\mathbf{Q}(0,0,0), \mathbf{R}(0,1,0), \mathbf{S}(4,0,2), \mathbf{T}(-1,2,3)$

Check back soon!

Problem 37

In Exercises 37 and 38, Determine whether or not the five points whose coordinates are given lie on a sphere.
$\mathbf{Q}(0,0,0), \mathbf{R}(2,-4,6), \mathbf{S}(3,-1,0), \mathbf{T}(4,-3,1), \mathbf{K}(-2,-1,5)$

Eduard Sanchez

Numerade Educator

Problem 38

Determine whether or not the five points whose coordinates are given lie on a sphere.
$\mathbf{Q}(0,0,0), \mathbf{R}(5,5,0), \mathbf{S}(-1,0,-1), \mathbf{T}(1,1,1), \mathbf{K}(4,6,-6)$

Eduard Sanchez

Numerade Educator

Problem 39

In Exercises 39-42, Identify the intersection of the two graphs whose equations are given. GRAPH CAN'T COPY.
$x^2+y^2+z^2-25=0 ; y-3=0$

Gregory Higby

Numerade Educator

Problem 40

Identify the intersection of the two graphs whose equations are given. GRAPH CAN'T COPY.
$x^2+y^2+z^2-25=0 ; x^2+y^2+(z-2)^2-16=0$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 41

Identify the intersection of the two graphs whose equations are given. GRAPH CAN'T COPY.
$x^2+y^2+z^2-25=0 ; x^2+y^2+(z-1)^2-16=0$

Heather Zimmers

Heather Zimmers

Numerade Educator

Problem 42

Identify the intersection of the two graphs whose equations are given. GRAPH CAN'T COPY.
$x^2+y^2+z^2-25=0 ; x^2+y^2+(z-1)^2-9=0$

AG

Ankit Gupta

Numerade Educator