5.1 introduction to locus 5.2 equations of straight lines 5.3 equations of circles 5.4 comparing...
TRANSCRIPT
5.1 Introduction to Locus
5.2 Equations of Straight Lines
5.3 Equations of Circles
5.4 Comparing Deductive Geometry and
Contents5 Locus
Coordinate Geometry
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5.1 Introduction to Locus
In Latin, the word ‘locus’ means place. Traditionally, locus is the path traced
out by a moving point that satisfies certain condition. In mathematics, locus is
the set of all points meeting some specified conditions.
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(a) Different forms of Equations of Straight lines
5.2 Equations of Straight Lines
A. Straight lines
This is usually called the point-slope form of the equation of a straight
line.
).( 11 xxmyy
The equation of the straight line having slope
M and passing through the point A(x1, y1) is
given by
(i) Point-slope form
Fig. 5.17
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5.2 Equations of Straight Lines
The equation of a straight line with y-intercept c
and slope m is
This is called the slope-intercept form of the
equation of a straight line.
y = mx + c.
Note that when (x, y) = (0, 0), the equation of the
above straight line passing through the origin
becomes
y = mx.
Fig. 5.18
Fig. 5.19
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12
12
1
1
xx
yy
xx
yy
(ii) Two-point form
5.2 Equations of Straight Lines
.12
12
xx
yyAB
of slope
Since AP and AB are on the same line, their
slopes must be equal. Thus
This is called the two-point form of the equation of a straight line.
If P(x, y) is any point on the line AB, then
.1
1
xx
yyAP
of slope
When two points given are A(x1, y1) and B(x2, y2), we have
Fig. 5.20
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5.2 Equations of Straight Lines
.1b
y
a
x
If a point P(x, y) is on a straight line with x-intercept a and y-intercept b, by
using the two-point form, we have
This is called the intercept form of the equation of a straight line.Fig. 5.21
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There are two special case we needed to pay attention to:
5.2 Equations of Straight Lines
In case of a horizontal line, the slope is zero.
The equation of a horizontal line is y = k.
In case of a vertical line, the slope is undefined.
The equation of a vertical line is x = h.
The x-axis is given by the equation y = 0 and the y-axis
is given by the equation x = 0.
Case 1:
Case 2: Fig. 5.22
Fig. 5.23
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5.2 Equations of Straight Lines
(b) Intersection of Two Straight Lines
For two straight lines on the same plane, they do not intersect each other if they are parallel, that is, having the same slope (see Figure 5.27(a)).
Two straight lines have one and only one point of intersection if the slopes of the lines are different. The coordinates of the intersecting point satisfy the two given equations. (see Figure 5.27(c)).
If the two straight lines overlap with each other, their equations are the same there will be infinitely many points of intersection (see Figure 5.27(b)).
Fig. 5.27(a)
Fig. 5.27(b) Fig. 5.27(c)
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From the above examples, the equations can be expressed in the form
5.2 Equations of Straight Lines
B. General Form of Equations of Straight Lines
which is called the general form of the equation of a straight line, where
A, B and C are constants.
1. A, B and C can be positive, zero or negative.
Ax + By + C = 0,
Notes:
2. The right hand side of the general form is zero.
3. In the general form of a straight line, A and B cannot both be zero.
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For an equation Ax + By + C = 0 (where B 0) of a straight line,
C. Features of Equations of Straight Lines
5.2 Equations of Straight Lines
If b = 0 but A 0, the general form becomes Ax + C = 0, that is
line. vertical a represents which,A
Cx
.B
Cy
B
A intercept- and slope
This straight line does not have y-intercept and the
slope of the straight line is undefined as illustrated in
Fig. 5.32. Fig. 5.32
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5.3 Equations of Circles
A. Circles
The locus of points having a fixed distance form a
fixed point in a plane is the equation of a circle
This is usually called centre-radius form of the
equation of a circle.
.)()( 222 rbyax
.222 ryx
The equation of a circle centred at the origin
becomes
Fig. 5.34
Fig. 5.35
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5.3 Equations of Circles
B. General Form of Equations of Circles
The equations of circles can be expressed in the form:
(where D, E and F are constants), which is called the general form of
equation of a circle.
Notes:
022 FExDxyx
2. The right hand side of the general form of a circle is zero.
3. In the general form of a circle, the coefficients of x2 and y2 are both
equal to one.
1. D, E and F can be positive, zero or negative.
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C. Features of Equations of Circles
5.3 Equations of Circles
have we circle a of equation an For ,022 FEyDxyx
.22
)2
,2
(22
FEDED
radius and centre
Remarks:
,022
22
F
ED If 1.
,022
22
F
ED If 2. the circle is wholly imaginary.
The circle is known as an imaginary circle.
the equation represents a circle of zero radius.
The circle reduces to a point and It is known as a point circle.
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5.4 Comparing Deductive Geometry and Coordinate Geometry
With the use of the coordinate system, problems in the deductive geometry can be tackled with the help of the coordinates and equations, using an analytical approach.