14.

PROBLEMS RELATING TO THE STRAIGHT LINE.

PROB. 1. To find the equation of a straight line which passes through a given point (a', y') and makes a given angle, 6, with the axis of x.

Let LL' be the line, and P' the point, the co-ordinates of which, referred to the axes OX, OY,

are (x', y'),

Through P' draw PF parallel to ox and produce CP', the ordinate of the given point, to G.

Let P be any other point in L L'. From P draw the ordinate PD, and let OD, PD, the co-ordinates of the

indeterminate point P, be denoted by x, y, respectively. Then the angle EP'G=w; EP'L=0; LP'G=P'PE=w—0; also PEPD-DE=PD-P′c=y—y';

But PE tan. PP'E X P′E = tan. 0× P′E = a′ × P′E, if the axes are rectangular;

Whence substituting for PE, P'E, their values,

a. If the angle is supposed variable, a' becomes a; whence y-y'a (x-x') is the equation of every straight line which can be drawn through the given point (x', y'). If the coordinates of the given point are (0, b'), the equation becomes y—b' = a' (x —0), or y = a'x+b'.

b. Since the equation of every straight line which passes through the given point (a', y') is y-y=a(x-x'); and, the co-ordinate axes being rectangular, the tangent of the angle which the straight line passing through the two given points

(x', y'), (x'', y'') makes with the axis of x is

y"-y'

x". -x

=a (Art.

6. a.); it follows that the equation of the straight line which passes through two given points is

The same expression denotes the equation of a straight line passing through two given points (x', y'), (x", y') when the co-ordinate axes are not at right angles to one another; y'-y'

but, in this case, the coefficient -Y' of (x−x') is equi

15. PROB. 2. To find the equation of a straight line which passes through a given point (x, y) and is parallel to a given straight line y" a' a"+b.

Since the required line passes through the point (a', y'), its equation is of the form y-y'a (x-x'). (Art. 14. a.)

Also since it is parallel to the given line y'a' x'' + b′, the angle which it makes with the axis of r is equal to the angle which the given line makes with the axis of x. (Euc. I. 29.) Whence a=a'

is the equation of the straight line which passes through the point (x, y), and is parallel to the given straight line y" = a' x' +b'.

16. PROB. 3. Let y = a' x + b', and y = a′′x + b′′ be the equations of two straight lines which cut each other it is required to find the co-ordinates of the point of intersection of these lines, and the angle which they make with each other.

Since the point in which the straight lines cut each other is common to both the lines, the values of the co-ordinates of that point are the same in the equations of both the lines. Whence, for the point of intersection, and for that point only, the y and the x of the equation y=ax+b' are simultaneously equal to the y and the r respectively of the equation y=a"x+b".

X

To obtain the expressions of these particular values of x, y, in terms of a', b', a", b", the given quantities in the equations of the two lines,

from yax+b' subtract y=a" x+b"
··. (a'—a') x=b'—b''.

Wherefore the co-ordinates of the point of intersection are

tan. '-tan. "

...tan. =tan. (6′ —0")=1+ tan. 6' x tan. 0 0'')

(Trigon.Art.37.)

(a)

=

If the straight lines are parallel, 0... tan. 0..a' =

a", as in Art. 15.

90°, tan. = ∞ .'.1+ a'a′′=0, and a′′=

-

1

It = This is the condition which must be satisfied, in order that two straight lines may be perpendicular to each other. If the first line is perpendicular to the axis of x, a' = ∞, and

= 0; a result from which it follows that the

or tan.

(a'-a") sin. w

'I + a'a'" + (a' + a'') cos. w

(b)

If the two straight lines are at right angles to each other: whence the denominator of the value

tan. tan. 90°

of tan. is equivalent to 0;

= 0.

If, also, the axes are rectangular, w = 90° and cos. w =

1

Whence, in this case a" =

as before.

a'

a. It follows, the co-ordinate axes being rectangular, that if the equation of a straight line is y=a'x+b', that of a straight line at right angles to it will be

17. PROB. 4. To find the

To find the equation of a straight line which passes through the given point (a', y') and makes a given angle, 4, with the given straight line y= ax + b'.

Since the required line passes through the point (x', y') its equation is of the form y-y'a (x-x') (Prob. 1. a.). The only quantity, in this equation, which it is necessary to determine is a.

If the axes are rectangular, a denotes the trigonometrical tangent of the angle which the required line makes with the axis of x.

If this line is L'c (see fig. of Prob. 3.), a=tan. L'CX=tan.0; LD being the given line, a' tan. LDX=tan. 0'.

Whence, also, CPD=0.

But L'CX=LDX — CPD or ◊ = '—

(Euc. 1. 32.)

(Trigon. Art. 37.)

or, denoting tan. 4 by t' and writing a, a' for tan. §, tan. 0', respectively,

Substituting this value of a in the equation y-y'a (x-x')

This is the equation of a straight line which passes through the given point (x', y'), and makes a given angle 4, the tangent of which is t', with the given straight line y=a'x+b': the axes being rectangular.

If L'C is perpendicular to LD,

90°, and tan. 4 or t'= ∞ .

‚(x ——-x'′).

(a)

This is the equation of a straight line which passes through a given point (x', y') and makes a right angle with a straight line whose equation is y=a'x+b'.

If the co-ordinate axes are oblique, a or a' denotes (Art. 14.) the ratio between the sines of the angles which the required line makes with the axes; and if this line is L'C, we shall have (Art. 9.)

tan. (=tan. L'CX)=

also tan. ' (tan. LDX)=

a sin. w
1+a cos. w

a' sin. w
1+ a' cos. w

Therefore the angle CPD being represented by as before, also a and now holding the places of a" and " respectively in the equivalents of tan. (Art. 16.), we obtain from the value of tan. at (b) in that Article,

α

tan. +a'a tan. ø + (a' + a) tan. cos. w=(a'-a) sin. w : hence

a {sin. w+(a+cos. w) tan. } =a' sin. w−(1+ a' cos. w) tan.

or

a=

a' sin. w-(1+ a' cos. w) tan. ¤•
(sin. w+(a+cos. w) tan.

It follows that, when the co-ordinate axes are oblique, the equation for a straight line which passes through a point (x', y'), and makes a given angle with the straight line y= a'x+b' is

a' sin. w- ·(1+a' cos. w) tan.
sin. w+(a'+cos. w) tan.

∞ and a becomes equal to

also

1+ a' cos. w

a' + cos. w

(x − x').