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CHAPTER IV.

STRAIGHT LINE CONTINUED.

65. WE have seen that each of the equations

Ax+By+C=0, A'x+B'y +C' = 0,

represents a straight line. We will now interpret the equa

tion

Ax+By+C+λ (A'x + By + C') = 0 .....

where X is some constant quantity.

I. Equation (1) must represent some straight line, because

it is of the first degree in the variables x, y. (Art. 16.)

II. The line represented by (1) passes through the intersection of the lines represented by

For the values of x and y which satisfy simultaneously (2) and (3) will obviously satisfy (1); that is, the point in which (2) and (3) intersect lies on (1).

III. By giving a suitable value to the constant λ the equation (1) may be made to represent any straight line which passes through the intersection of (2) and (3).

For let x, y, denote the co-ordinates of the point of intersection of (2) and (3); suppose any line drawn through this point, and let x, y, be the co-ordinates of another point in it. Now we have already shewn in II. that the line (1) passes through (x, y); we have therefore only to prove that by giving a suitable value to λ the line (1) can be made to pass

through (x, y), because two straight lines which have two common points must coincide. Substitute Y2 for x and y respectively in (1), and determine A so as to satisfy the equation. Thus

Now use this value of λ in (1); then the equation

Ax+ By +C_ Ax, +By, + C

2

A' x12+ B'y2 + C2 (A' x + B'y + C') = 0... (4)

represents a straight line passing through (x,, y1) and (x, y).

We have thus proved that by giving a suitable value to λ, the equation (1) will represent any straight line passing through the intersection of (2) and (3).

66. The preceding article is very important, and commonly presents difficulties to beginners. The student should not leave it until he is thoroughly familiar with the three propositions which are contained in it. The first proposition is obvious. To prove the second proposition the student may, if he pleases, actually find the values of x and y which satisfy simultaneously Ax+By+C=0, and A'x + B'y + C' = 0, and convince himself, by substituting these values, that they do satisfy Ax+ By +C+λ (A'x + B'y + C') = 0. There is, however, no necessity for solving the first equations, because it is evident that values of x and y which make Ax+By+ C and A'x + B'y +C' vanish simultaneously must make Ax+ By +C+λ (A'x+ B'y +C') vanish, because they make each of the two members of the expression vanish. The third proposition of the preceding article is usually the most difficult the student is apt to think it needs no demonstration. It may be obvious, however, that by giving different values to λ, different lines are represented, and that we can thus obtain as many lines as we please, but this does not shew that we can by a suitable value of A in (1) represent any line passing through the intersection of (2) and (3).

For example, if the straight lines (2) and (3) be DSE and FSG respectively, it might have happened that all the lines represented by (1) fell within the angle FSD and none

within FSE. It requires to be proved then that by giving to λ a suitable value in (1) we can obtain the equation to any line through S.

x y

a

67. It is often convenient to denote by a single symbol the expression which we equate to zero in our investigations in this subject; for example, in Art. 51 we have used the symbol a as an abbreviation for x cos ay sina - p. In like manner we may denote such expressions as Ax+By+C, y-mx-c, 1,... by single symbols, as u, v,... u',... Now it will be seen that the demonstration in Art. 65 applies to any form of the equation to a straight line as well as to the form Ax+By+C=0 which we have used. Hence the result may be enunciated thus:-if u = 0 and v = 0 be the equations to two straight lines, and λ a constant quantity, the equation u+λv = 0 will represent a straight line passing through the intersection of the two lines; and by giving a suitable value to λ, the equation will represent any straight line passing through the intersection of the two lines.

68. If u=0 and 0 be the equations to two straight lines, then as we have shewn, u+v=0 will represent a straight line passing through their intersection; it is sometimes convenient to use the more symmetrical form lu+mv=0, where 7 and m are both constants. It is obvious that what has been said respecting the first form applies to the second; in

m

fact the second is deducible from the first by writing for λ. Τ It must be remembered throughout this chapter that 7, m, n,...,... are constants, though for shortness we may omit to state it specially in every article.

69. Similarly if u =0, v=0, w=0, be the equations to three straight lines, and l, m, n be constants, the equation lu+mv + nw = 0........

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will represent a straight line. Moreover, by giving suitable values to l, m, n we may in general make this equation represent any straight line whatsoever. For suppose we wish this equation to represent the straight line passing through (x, y) and (x,, y1). Let u1, v1, w, denote the values of u, v, w respectively when we put x, for x and y, for y; and let u,, v,, w, be the respective values when x, and y, are put for x and y respectively. Then determine the values of and Τ from the equations

m

n

which represents the line passing through the points (x, y) and (x,, Y2).

We have said above that the equation (1) can in general be made to represent any straight line, because there are exceptions which we now proceed to notice.

When the lines represented by u=0, v=0, and w=0 meet in a point, the equation (1) represents a line which necessarily passes through that point. For since the three given lines meet in a point, u, v, and w vanish simultaneously at that point; therefore lu+mv+nw also vanishes at that point, so that the line represented by equation (1) passes through that point.

When the three given lines are parallel the equations u = 0, v = 0, w = 0 will be of the form

Ax+By+C=0,

Ax+By+C2 = 0,

Ax+By+ C1=0,

and thus equation (1) may be reduced to

and this equation represents a line parallel to the given lines.

Thus if the three given lines meet in a point or are parallel, equation (1) will not represent any straight line; for the line represented by equation (1), in the former case passes through the point in which the given lines meet, and in the latter case is parallel to the given lines.

We may shew that there is no other exception. For the only case in which the general investigation can fail is when λ, μ, and v all vanish, that is, when

We shall now prove that when equations (2) are satisfied, the three given lines either all meet in a point or are parallel.