Ex. 1. To find the intersection of the lines whose equations are

=

y=3r+ 1 and y 2x-40. X 3 and Y = 10. Ex. 2. To find the intersection of the lines whose equations are

If a third line, whose equation is y = ax + b", passes through the point of intersection, the relation between the coefficients is

(a b'a' b) - (a b'' — a'' b) + (a'b'' — a'b') = 0.

45. To find the tangent, sine and cosine of the angle betwen two given straight lines.

Let yax+b be the equation to C B

Ø and ' the angles which they make respectively with the axis of r; then

46. To find the equation to a straight line making a given angle with

Substituting this value for a' in the second equation,

If the required line passes also through a given point x, y, the equa tion is

α- β
1+aß (x − x1).

If D be considered the given point x, y, then not only the line DOE but another (the dotted line in the figure) might be drawn, making a given angle with B C, and its equation is found, as above, to be

For example, the two straight lines which pass through the point D and cut B C at an angle of 45° are given by the equations

Also the equation to the straight line passing through D and cutting the axis of x at an angle of 135° is

47. If the required line is to be perpendicular to the given line, 6 is

infinitely great; therefore the fraction

; hence the equation to a straight line perpendicular to a

This may be also directly proved, for drawing OE perpendicular to B C, as in the next figure, we have a

tan. O EX=

hence in the equations to two

=

tan. OEC =

straight lines

0; and, con

which are perpendicular to one another we have a a' + 1 versely, if in the equations to two straight lines, we find a a' + 1 = 0, these lines are perpendicular to one another.

If the perpendicular line pass also through a given point 1 y1, its equation is

and, of course, this equation will assume various forms, agreeing with the position of the point x, y1; thus, for example, the line drawn through the origin perpendicular to the line y ax + b, is one whose equation is

48. To find the length of a perpendicular from a given point D (x, y) on a given straight line C B.

The superior sign is to be

taken when the given point is above the

given straight line, and the inferior in the contrary case.

απι

If the given line pass through the origin b=0; .. p= ± Y1 = α x1

If the origin be the given point, a1 = 0 and y1 = 0; :p=

There is another way of obtaining the expression for p. Since the equation y = ax + b applies to all points in C O B, it must to Q, where M D or y, cuts CO B ; .. MQ — « x1 + b; Now DO DQ sin. DQ O, =

and sin. DQO sin. CQM= cos. QCM =

49. If the line D E had been drawn making a given angle whose tangent was ẞ with the given line C O, the distance DO might be found; for instead of equation (2) we shall have

hence, following the same steps as above, we shall find

This expression is also very easily obtained trigonometrically.
Let y sine of the given angle, then

50. The equation to the straight line may be used with advantage in the demonstration of the following theorem :

D

L

B

x1

(x-x) (47)

(x-x2) (41)

X1 X2

or y=

(xx) since y2- 0.

for the intersection O of B D and A E we have, by equating the values of y,

X1 X2

X,

X1 X2 X1 X · X2 X;:. X2 X = x1 x and x = x12

that is, the abscissa of the point O is found to be that of C.

In the same manner it may be proved that if perpendiculars be drawn from the bisections of the sides, they will meet in one point.

Similarly we may prove that the three straight lines FC, K B, and A L, in the 47th proposition of Euclid, meet in one point within the triangle A B C.

51. We have hitherto considered the axes as rectangular, but if they be oblique, the coefficient of x, in the equation to a straight line, is not the tangent of the angle which the line makes with the axis of

Let

then α =

the angle between the axes,

X.

the angle which the line makes with the axis of æ ;
y-b

sin. 0

Ꮖ sin. (w-0)

(33);

b remains, as before, the distance of the origin from the intersection of the line with the axis of y: hence the equation to a straight line referred to oblique axes is

Since this equation is of the form yax+b all the results in the

preceding articles which do not affect the ratio of

equally true when the axes are oblique.

sin.

will be

Thus, articles 40, 41, 42, 43, and 44, require no modification.
To find the tangent (B) of the angle between two given straight lines.

To find the equation to a straight line passing through a given point x y1, and making a given angle with a given straight line.

Let ẞ be the tangent of the given angle,

yax+b, the given line,