1969-05-08 — Page 26

袁二第張七第日二廿月三年巴夏 WAH KIU

PO

1969

現代數學科

(廿七) 王錦釗,

MODERM MATHEMATICS (27

Solutions of the exercises assigned last week.

(1). Sketch the graph of

The graph of the given system is the triangular region OAB including the boundary.

(2)Sketch the graph of (3x+ys 12.

3x-2y-6

x + y 2 ix - 2y6.10

報日橋

Courdinate geometry of one dimension

四期星日八月五年九六九一醚公年八十五圃民塞中青教僑

Coordinate geometry of one dimension 18 established by associating points in a line, called number line, with numbers. If the universe is the set of real numbers D, then for every real number there is a corresponding point in the line. and for every point in the number line there is a corresponding number. In other words the system

74-451-1-1954142 43 44 45 +6 +7 +8 establishes a one-to-one correspondence. De ween a real number and a point in the line. We say the point is in the line, because a line is a set of points and a point is in the set. The number associated with the point is called the. coordinate of the point. In the figure above, thn coordinate of A 18 (+3), that of 8 is (+6), and that of C and D are (-2) and (-5) respectively.

Given a condition in one variable, the graph of the condition is the set of all points in the line such that the coordinates of the points are members in the solution set of the condition. The graph of a condition in one variable may be s point, or a Bet of points. The set of points may be a ray, or a half-line, or a segment, and so on.

капріев

(1) Find the solution set of x, universe

being the set of real numbers. then draw its graph,

Solutions

Solution Bet

Graphi

3

2.

0

+4 +3 The graph is a ray which contains all the points whose coordinates are +3 or greater than +3

+1

+5, +6 +7

Examplet⠀

The distance between A(3,4) and B1-1;-5)

(~5-4)?

√16 + 81

√97

(3) Division formula

Theorem. If P (x,y) davides

日

12

1030 2

Begmente such that a PP,

I

F. then

(x-1) and yy.

ין

Let M1, Mm M2 be the orojections of P. Fanta

The x-axis; and N. he the

onto y-ax18. projections of P... ► "2

F, and

x=dy 10

(2) Find the solution set and the graph of

<,universe being the set of real

numbers,

Solution

Solution set xxx < + {

The graph is the region ABCD including the boundary.

(3) A rigor manager is going to install two types of

maobines, a small and a large. The table shows the restricting conditions. The profit on a small machine is £3 and on a large one i8 £5. Find the maximum profit in a week.

(0,17)

(28,0)

No. of operations Space in yard:

Small

2

Large

3

8

Mex,

available

56

136

Solution

-System of constrainta

12x + 3y $ 56

4x + 8y $ 116

ly 20

Graph:

**

-4

-2

+1 +2

The graph is a half-line not including the end point whose coordinate is (-1). (3) Find the solution set and graph of

14 and 1-2. universe being the set of real numbers,

PP1

>Solution

Solution sat

Graph

-4 -3 -2

B

+2 13

+

OF

N

い

1.

M

T:

The graph is the segment AB.

(4) Find the solution set and graph of 1 § - or 1-1, universe being the set of real numbers.

solution 1 -

Solution set. - 3/25 −1 or

Grapht

the universe

Or I • I. • г (x,

Similarly,

y = yr (y2

Examples I

-

(a) Find the coordinates of the point which trisect the segment joining ▲(2,3) and B(7,12).

Solution

Let P(x,y) be the dividing point,

4-3-2

a +! +2 +3 +4 m

The graph is the number line itself.

3. Coordinate geometry of two dimensions

(1) Introduction

We are to use two perpendicular lines to Locate points in the plane. The vertical line Le called the y-axis whereas the horizontal, line is called the x-axis. Let D x D be the

eet of all ordered pairs of real numbers. Then there is a one-to-one correspondence, between. points in a plane and the ordered pairs in Dx D. The intersection of the two axes is called the origin, denoted by 0. Given a point P in the plane, we drop a perpendicular to x-axis meeting it at M, The coordinate x of M in x-axis is called the x-coordinate. We drop a perpendicular from ? to y-axis, meating it at N. The coordinate y of N in y-axis is called the y-coordinate. Thus to every point P in the plane there corresponds an ordered pair (x,y) of real numbers.

I AB

1

}

I - 2+

(7

2)

y

5+

- 2 + ог

}(12 -

• 1 * 3 OF

"}

? 18 at (3, 6).

(b) Find the coordinates or the point which

bisects the segment joining Al -2,-4) and B(6,14).

Solution.

Let P(x,y) be the point

AP

ED

AB

Obiective: f(x,y) = 3x - .5x

At B (10,12), f(x,y) 1ẻ greatest

e (10:12) 10 x 3 + 12 x 5 90

XIII Coordinate geometry in one and two

Dimensions

1. introduction

In algebra, we consider sets of numbers, and in geometry we consider sets of points. In algebra, we learn how to solve conditions, that are linear, quadratic, cubic and so on. These conditione may be equations or inequations of one or two variables. To solve a condition is to find the set of all numbers or all ordered pairs of numbers that satisfy the given condition. In geometry, we say that any set of points is a geometric figure. We study the properties of different sets of points from the metric point of view as well as the non-metric point of view. Nowadays we wouldn't treat the subjects of algebra and geometry isolatedly. Since both. algebra and geometry can be interpreted and understood in terms of sets, we can integrate the two subjects as a whole under the topic of what we call coordinate geometry. Since in algebra we only consider equations and inequations of one two variables, at present we only deal with coordinate geometry in one or two dimensions "accordingly.

OF

2) Distance formula

P(X.Y

→

M

Theorem: The distance between the points

(x, y) and F2(ką, y2) is given by the formula

3ï32 - √(x2 − ×,)2 + (32

-

FD

where

means

2

the measure or length of the segment PF2.

FRODIT, LAT X4: My Agi #2 be the projections or

and P2 onto the axes. Then

n P.P

P1P2 N

·

نے سے

*

#16 - 2.

4 or 2

- -4

+

(14 + 4)

--4

*

9 or 5

Pis at (2,5).

4. Exercises for this week:

(1) Locate the following pointe in the coordinate

plane.

(-5,3); (-3,-4); (7,0); (0,-61; (4,-8)1(5,7) (2) Find the distance between each pair of the

following pointer

(-9,-3), (5,0); (4,-6); (0, -21; \ ~ $2, -7), 10,3).

(3) Find the coordinates of the point P(x,y) such

that it bisects the edgment joining A(~4,0) and 8(0,12).

14) Given P.(-6,8), P2(16,24), find the

coordinates of P such that a PP ว

• -2: 5

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