Chapter 18 Locus

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5.1 Introduction to Locus 5.2 Equations of Straight Lines 5.3 Equations of Circles 5.4 Comparing Deductive Geometry and 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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5.2 Equations of Straight Lines A. Straight lines (a) Different forms of Equations of Straight lines (i) Point-slope form The equation of the straight line having slope M and passing through the point A(x1, y1) is given by y − y1 = m( x − x1 ). Fig. 5.17 Content

This is usually called the point-slope form of the equation of a straight line.

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5.2 Equations of Straight Lines Note that when (x, y) = (0, 0), the equation of the above straight line passing through the origin becomes y = mx. Fig. 5.18

The equation of a straight line with y-intercept c and slope m is y = mx + c. Content

This is called the slope-intercept form of the equation of a straight line.

Fig. 5.19

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5.2 Equations of Straight Lines (ii) Two-point form When two points given are A(x1, y1) and B(x2, y2), we have

slope of AB =

y2 − y1 . x2 − x1

If P(x, y) is any point on the line AB, then slope of AP =

y − y1 . x − x1

Since AP and AB are on the same line, their slopes must be equal. Thus Content

Fig. 5.20

y − y1 y2 − y1 = . x − x1 x2 − x1

This is called the two-point form of the equation of a straight line.

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5.2 Equations of Straight Lines 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

x y + = 1. a b

Fig. 5.21

This is called the intercept form of the equation of a straight line. Content

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5.2 Equations of Straight Lines There are two special case we needed to pay attention to: Case 1: In case of a horizontal line, the slope is zero. The equation of a horizontal line is y = k. Case 2:

Fig. 5.22

In case of a vertical line, the slope is undefined. The equation of a vertical line is x = h. Content

The x-axis is given by the equation y = 0 and the y-axis is given by the equation x = 0.

Fig. 5.23

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5.2 Equations of Straight Lines (a)

Intersection of Two Straight Lines

Fig. 5.27(a)

Fig. 5.27(b)

Fig. 5.27(c)

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)). 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)). Content

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)).

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5.2 Equations of Straight Lines B. General Form of Equations of Straight Lines From the above examples, the equations can be expressed in the form Ax + By + C = 0, which is called the general form of the equation of a straight line, where A, B and C are constants. Notes : 1. A, B and C can be positive, zero or negative. Content

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The right hand side of the general form is zero.

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In the general form of a straight line, A and B cannot both be zero.

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5.2 Equations of Straight Lines C. Features of Equations of Straight Lines For an equation Ax + By + C = 0 (where B ≠ 0) of a straight line, slope = −

A C and y - intercept = − . B B

If b = 0 but A ≠ 0, the general form becomes Ax + C = 0, that is x=− Content

C , which represents a vertical line. A

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 ( x − a 2 ) + ( y − b) 2 = r 2 . This is usually called centre-radius form of the equation of a circle.

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Fig. 5.34

The equation of a circle centred at the origin becomes x2 + y 2 = r 2. 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: x 2 + y 2 + Dx + Ex + F = 0 (where D, E and F are constants), which is called the general form of equation of a circle. Notes:

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D, E and F can be positive, zero or negative.

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The right hand side of the general form of a circle is zero.

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In the general form of a circle, the coefficients of x2 and y2 are both equal to one.

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5.3 Equations of Circles C. Features of Equations of Circles For an equation of a circle x 2 + y 2 + Dx + Ey + F = 0, we have 2

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D E  D  E centre = (− ,− ) and radius =   +   − F . 2 2  2 2 Remarks : 2 2  D  E 1. If   +   − F = 0, the equation represents a circle of zero radius.  2 2 The circle reduces to a point and It is known as a point circle. Content

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 D  E 2. If   +   − F < 0, the circle is wholly imaginary.  2 2 The circle is known as an imaginary 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.

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