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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