10.6 Quadric Surface Notes

10.6 Quadric Surface Notes - 1 0 .6 

Q u a d r ic 
S...

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Q u a d r ic 
S u r fa c e s 
 
 I. Cylinders:

Any
equation
that
contains
only
2
of
the
3
 variables.

(The
third
is
unrestricted.)

 
 x 2 + z 2 = 4 





Sketch
the
cross
sections
for

 Ex.
 y
=
constant.
 
 
 
 
 
 
 
 
 
 
 
 yz = 1 Ex.
 







Sketch
the
cross
sections
for

 y
=
constant.
 
 
 
 
 
 
 
 
 y = x 2 




 Do:

Sketch
 Sketch
the
cross
sections
for

 y
=
0
and
y
=
4.
 
 
 
 
 II
Quadric
Surface
–
Graph
of
2nd
degree
equation
in
3
 variables
x,
y,
and
z.
 
 A. All
3
variables
squared.
 1. Ellipsoid:

3
positive
coefficients
 
 
 x 2 y2 z 2 + 2 + 2 =1 2 a b c 






(a
=
b
=
c:

sphere)
 x 2 y2 + + z2 = 1 9 Ex.
 16 
 
 
 
 
 
 
 
 
 
 
 2. Hyperboloids:
 a. One
sheet:

one
negative
coefficient
 
 9 x 2 + 36 z 2 ! 4 y 2 = 36 
 Ex.
 
 
 
 
 
 
 
 
 
 
 
 b. Two
sheets:

two
negative
coefficients
 z2 y2 ! x 2 ! = 1 4 Ex.
 
 
 
 
 
 
 
 ! x 2 + y 2 ! z 2 = !1 
 Ex.
 
 
 
 
 3. Cones:

(one
or
two
negative
coefficient;
variables
set
=
 0)
 
 x2 z2 2 +y ! =0 9 Ex.
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 B. Paraboloids
(2
variables
squared)
 
 1. Elliptic
paraboloid
 x 2 y2 + =z 4 9 Ex.
 
 
 
 
 
 
 
 
 
 2. Hyperbolic
paraboloid
 z = y2 ! x 2 
 Ex.
 
 
 
 
 
 
 Do:

Sketch
the
following:
 1. 
 x 2 + y2 + z 2 = 4 
 ! x + y + z = 4
 2 2. 
 
 2 2 ! x 2 ! y2 + z 2 = 4 
 3. 
 
 x 2 ! y2 + z 2 = 0
 4. 
 
 x 2 ! y2 = 4 
 5. 
 
 x=y +z 2 6. 
 
 
 
 
 
 
 2 
 ...
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