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ANALYTIC GEOMETRY
Math 14
Plane and Analytic Geometry
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SPACE COORDINATES
andSURFACES
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OBJECTIVES:
At the end of the lesson, the student is expected to be
able to:
define space coordinates.
plot points of space coordinates.
write and sketch the graphs of space coordinate
equations.
know the different kinds planes and surfaces.
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Let OX, OY, and OZ be three mutually perpendicular
lines. These lines constitute the x-axis, the y-axis, and the
z-axis of a three-dimensional rectangular coordinatesystem. The axes, in pairs, determine three mutually
perpendicular planes called coordinate planes. The planes
are designated as the XOY-plane, the XOZ-plane, and the
YOZ-plane or, more simply, the xy-plane, the xz-plane, andthe yz-plane. The coordinate planes divide space into eight
regions called octants. The distance of P from the yz-plane
is called the x-coordinate, the distance from the xz-plane
the y-coordinate, and the distance from the xy-plane the
z-coordinate. The coordinates of a point are written in the
form (x, y, z), in this order, x first, y second, and z third.
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z
x y
o
xy-plane
yz-planexz-plane
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Example:
Plot the given points in a three-dimensional
coordinate system.
1. (3, 0, 0)
2. (0, 3, 0)
3. (0, 0, 3)
4. (1.5,-1, 2)
5. (0, 2, -2)
6. (2, 2.5, 3)
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(0, 0, 3)
(0, 3, 0)
(3, 0, 0)
(2, 2.5, 3)(1.5,-1, 2)
(0, 2, -2)
z
x
yo
-z
-x
-y
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THEOREMS:
Let P1(x1, y1, z1) and P2(x2, y2, z2) be the coordinates of
two points in a three-dimensional coordinate system. Thenthe distance d between P1 and P2 is given by
The coordinates P(x, y, z) of the midpoint of the line
segment joining P1(x1, y1, z1) and P2(x2, y2, z2) are given by
the equations
This theorem may be generalized by letting P(x, y, z) beany division point of the line through P1 and P2. If the ratio
ofP1P to P1P2 is a number r, then
212
2
12
2
12zzyyxxd
212121
2
zzz
2
yyy
2
xxx
121121121zzrzzyyryyxxrxx
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EXAMPLES:
1. Find the distance between the points P1(-4, 4, 1) and
P2(-3, 5,-4).2. Find the coordinates of the midpoint of the line
segment that joins A(3,-2, 4) and B(-6, 5, 8).
3. Find the coordinates of the point P(x, y, z), which is
one-third of the way from A(1, 3, 5) to B(5, 7, 9).
4. Given: A(1, 4, 7) and B(5,-1, 11), find the point P so
that the ratio of AP to PB is equal to 4 to 7.
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SURFACES
A. PLANE: Ax + By + Cz + D = 0
a) x = k, plane parallel to yz-plane
b) y = k, plane parallel to xz-plane
c) z = k, plane parallel to xy-plane
d) Ax + By + D = 0, plane parallel to z-axise) By + Cz + D = 0, plane parallel to x-axis
f) Ax + Cz + D = 0, plane parallel to y-axis
g) Ax + By + Cz = 0, plane
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EXAMPLE:
1. x = 3 2. y = 3
y
x
z
y
x
z
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3. z = 3
y
x
z
y
x
z
(0, 4, 0)
(6, 0, 0)
4. 2x + 3y = 12
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5. 2x + 3z = 12
y
x
z
y
x
z
6. 2y + 3z = 12
(0, 0, 4)
(6, 0, 0)
(0, 6, 0)
(0, 0, 4)
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6. 2x + 3y + 4z = 12
y
x
z
(0, 4, 0)
(0, 0, 3)
(6, 0, 0)
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B. CYLINDERS and SPHERE:
1. x2 + y2 = 4
y
x
z
(-2, 0, 0)
(0, 2, 0)
(2, 0, 0)
(0,-2, 0)
CIRCULAR CYLINDER
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2. 4x2 + y2 = 4
y
x
z
(-1, 0, 0)
(0, 2, 0)
(1, 0, 0)
(0,-2, 0)
ELLIPTICAL CYLINDER
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3. x2 = y
y
x
z
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4. y2 = x
y
x
z
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5. z2 = y
y
x
z
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5. z2 = y 1
y
x
z
V(0, 1, 0)
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SPHERE: (x h)2 + (y k)2 + (z l)2 = r2
Ax2 + Ay2 + Az2 + Gx + Hy + Iz = J
r = 0 (point)
r = - (no locus)
r = + (sphere)
y
x
z
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EXAMPLE:
Describe the locus of x2 + y2 + z2 + 2x 4y 8z + 5 = 0.
Sketch the graph.
SOLUTION:
x2 + 2x + 1 + y2 4y + 4 + z2 8z + 16 = 5 + 1 + 4 + 16
(x + 1)2 + (y 2)2 + (z 4)2 = 16
C(1, 2, 4) and r = 4
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y
x
z
y
x
z
(-1, 6, 4)
(-5, 2, 4)
(-1, 2, 0)
(3, 2, 4)
(-1,- 2, 4)
(-1, 2, 8)
C(-1, 2, 4)
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QUADRIC SURFACESSecond-Degree Equation in x, y, z
Form: Ax2+By2+Cz2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0
The graph of such equation is called quadric surface or
simply quadric.
Six Common Types of Quadric Surfaces:
1. Ellipsoid
2. Hyperboloid of One Sheet
3. Hyperboloid of Two Sheets
4. Elliptic Paraboloid
5. Hyperbolic Paraboloid
6. Elliptic Cone
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QUADRIC SURFACES
ELLIPSOID
1
c
z
b
y
a
x2
2
2
2
2
2
x
y
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QUADRIC SURFACES
HYPERBOLOID OF ONE SHEET
1
c
z
b
y
a
x2
2
2
2
2
2
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QUADRIC SURFACES
HYPERBOLOID OF TWO SHEETS
1
b
y
a
x
c
z2
2
2
2
2
2
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QUADRIC SURFACES
ELLIPTIC PARABOLOID
2
2
2
2
b
y
a
xz
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QUADRIC SURFACES
HYPERBOLIC PARABOLOID
2
2
2
2
a
x
b
yz
x
y
z
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QUADRIC SURFACES
ELLIPTIC CONE
2
2
2
22
b
y
a
xz
EXAMPLES: Sketch the quadric surface
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EXAMPLES: Sketch the quadric surface.1. 36x2+9y2+4z2=36
Solution:
19
z
4
y
1
x
36136z4y9x36
222
222
x y z
x 1 0 0
y 0 2 0
z 0 0 3
:Intercepts.I
(ellipse)
14
y
1
x:0zlet
plane-xyi)
:Traces.II
22
(ellipse)
19
z
4
y:0xlet
plane-yziii)
22
(ellipse)
19
z
1
x:0ylet
plane-xzii)22
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y
x
z
(-1,0,0)
(1,0,0)
(0,-2,0) (0,2,0)
(0,0,-3)
(0,0,3)
14
y
1
x22
19
z
4
y 22
19
z
1
x22
2 16x2+36y2 9z2=144 plane-yzii)
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2. 16x2+36y2-9z2=144:Solution
116
z
4
y
9
x
144
1144z9y36x16
222
222
:Intercepts.I
x y z
x 3 0 0
y 0 2 0
z 0 0 4i
(ellipse)
14
y
9
x:0zlet
plane-xyi)
:Traces.II
22
)(hyperbola
116
z
4
y:0xlet
planeyzii)
22
)(hyperbola
116
z
9
x:0ylet
plane-xziii)
22
(ellipse)
18
y
18
x
2
12
4
y
9
x
114
y
9
x
116
)4(
4
y
9
x
4zlet
:plane-xytoparallelSectionsIII.
2222
22222
z ( 4 2 0 4)22
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y
x
z
y
y
x
x
(-4.2,0,-4)
(3,0,0)
(0,2,0)(0,-2,0)
(-3,0,0)
(0,2.8,-4)
(4.2,0,4)
(0,2.8,4)(0,-2.8,4)
(-4.2,0,4)
(0,-2.8,-4)
(4.2,0,-4)
z=4
z=-4
18
y
18
x22
14
y
9
x22
116
z
9
x22
116
z
4
y22
3 4z2 4x2 y2=4 plane-yzii)
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3. 4z -4x -y =4:Solution
14
y
1
x
1
z
4
14yx4z4
222
222
:Intercepts.I
x y z
x i 0 0
y 0 2i 0
z 0 0 1
trace)(no
14
y
1
x:0zlet
plane-xyi)
:Traces.II
22
)(hyperbola
14
y
1
z:0xlet
py)
22
)(hyperbola
11
x
1
z:0ylet
plane-xziii)
22
(ellipse)
132
y
8
x
8
18
4
yx
14
y
1
x914
y
1
x
1
3)(
3zlet
:plane-xytoparallelSectionsIII.
2222
22222
z22
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y
x
z
y
y
x
x
(-2.8,0,-3)
(0,0,-1)
(0,0,1)
(0,-5.7,-3)
(0,5.7,3)
(0,5.7,-3)
(2.8,0,3)
(-2.8,0,3)
(0,-5.7,3)
(2.8,0,-3)
z=3
z=-3
132
y
8
x22
14
y
1
z22
1xz22
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REFERENCES
Analytic Geometry