Cylinders And Quadric Surfaces Worksheet With Answers
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313.6 Cylinders and Quadric Surfaces
Cylinders Graphs in xyz space whose equation is f (x, y) = C, f (y, z) = C or f (x, z) = C.
2
f (x, y) = C
draws a curve in
(Remark)
3
f (x, y) = C
draws a surface in
2
2
2
3
Example Sketch x
+ y
= 1 in
and in
respectively.
2
2
3
Example Sketch y
= 1 in
.
z
Quadratic Surfaces A graph in xyz space whose equation is a second-degree polynomial in x, y, and z is called a quadratic
surface. The most general form of an equation for a quadratic surface us
2
2
2
Ax
+ By
+ Cz
+ Dxy + Eyz + F xz + Gx + Hy + Iz + J = 0
where A, B,
, J are constants, and at least one of A, B, and C is nonzero.
Remark By rotations and translations, the equation for a quadratic surface can be written in one of the two
forms below :
2
2
2
Ax
+ By
+ Cz
+ J
= 0
(at least one of A, B, and C is nonzero.)
or
2
2
(At least one of A and B is nonzero.)
Ax
+ By
+ Iz
= 0
Definition (Trace or cross-section) The intersection of a surface f (x, y, z) = C with a plane parallel to a coordi-
nate plane is called a trace or a cross-section of the surface.
2
2
2
(e.g.) Let
be the quadratic surface whose equation is x
+ y
+ 4z
= 4.
2
2
2
When
is cut by x = k
:
y
+ 4z
= 4
k
2
2
2
When
is cut by y = k
:
x
+ 4z
= 4
k
2
2
2
When
is cut by z = k
:
x
+ y
= 4
4k
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