This is the equation for a straight line which passes through a given point (x', y') and makes a right angle with the straight line y-ax+b'; the axes being oblique.

18. PROB. 5. Given the equation of a straight line, y=ax+b', and a point (x, y) without it, to find the length of the perpendicular drawn from the given point to the given straight line.

The equation of the straight line which passes through the point (x', y') and makes right angles with the straight line y=a'x+b', is

y—y' = —

y-y=

(Prob. 4.)

1

a

(x-x), if the axes are rectangular, and

1+ a' cos. w

a' + cos. w

(x-x'), if the axes are oblique

1st. Let the axes be rectangular.

Then x', y' being the co-ordinates of one extremity of the line whose length is required, if the values of x, y,, the coordinates of the point of intersection of the two lines are determined, and the differences x-x', y,-y' are substituted for x- and y-y" in the formula D= √ {(x' — x'')2 + (y'--y'')2} (Art. 6.), the result will be that which is required.

Since x, y, the co-ordinates of the point of intersection, are common to both lines, the equations of the lines for that point are

Multiplying the terms of eq. 1. by a', the terms of eq. 2. by a', and adding the results,

Substituting this value of x,- in equation 2.

y,—y' = —

1 a' (y' — a'x' —b′),
X a2+1

a

y' — a'x' — b'

a' 2+1

Substituting these values of x,-x', y,—y' in the formula which expresses the distance between two points, in terms of the co-ordinates of these points, it is found that the length of

C

the perpendicular from the point (x', y') on the straight line y=a'x+b', is

As the value of D is positive, it is necessary to take the upper sign when the numerator y'-'x'-b' is positive, and the lower when it is negative.

If the point (x', y') coincides with the origin, x'=0, y′ =0, and

Putting the equation of the given line, y,=a'x,+b', under the form

y-y'a' (x,-x') + b' + a' x' —y',

and combining it with the equation

Substituting these values of x,-x', y,-y' in the formula, D= √ {(x, −x')2+(y, −y')2 + 2(x, − x') (y,-y') cos. w}(Art.7.) and making the necessary reductions, it is found that

(y'—a'x' —b′) sin. w D= ± √ {1+a22+2 a' cos. w}"

CHAP. III.

TRANSFORMATION OF CO-ORDINATES.

19. THE general equation of a straight line is y-ax+b, and the equation of a straight line which passes through the origin of the co-ordinates, y = a a ; the quantity denoted by a in both equations being different, according as the line is referred to rectangular or oblique axes, and more complicated in the case of the latter.

It is hence evident that the same line may be represented by an equation which is more or less simple, according as its position with respect to the axes is more or less simple, and also according as the axes are rectangular or oblique.

Consequently, if the position of a line on a plane is already determined by means of an equation, and it is observed that this line is in a more simple situation, with respect to two new straight lines than with respect to the primitive axes, it becomes of importance to deduce the equation of the line referred to the new axes, from the equation of the line referred to the primitive axes. This is the object proposed in the transformation of co-ordinates. The problem will be resolved if, for any point of the line, there are known the values of the primitive co-ordinates in functions of the new; for if these values are substituted in the equation proposed, the result gives a relation between the new co-ordinates of each of the points of the line under consideration.

Let ox, or be the primitive axes, and oc = a, PC=y, the co-ordinates of any point P, referred to these axes: also let 0,X,, O,Y,, be the new axes, and let o, Ex,, PE=y,, be the co-ordinates of the same point P, referred to the new axes.

Through o, let Bo, y' be drawn parallel to or, and ox” parallel to ox; also through E let EGH be drawn parallel to o r, and E F parallel to ox.

Then there are given OB-x', O,B=y', the co-ordinates of o,, the new origin;

xor = w, the angle made by the primitive axes ;

x"o, x,=w', the angle made by the primitive and the new axes of x; and

X"0,Y=", the angle made by the primitive axis of x and

the new axis of y; and that which is required is to express r and y, the primitive co-ordinates of any point P, in functions of the given quantities x', y', w, w', w'', and of x, y,, the new co-ordinates of the point P.

Because ox, 0,X", EF are parallel lines, and OY, BY", HE and CP are also parallel lines,

X"0,Y"=XOY=w.

Y

114

E

F

G

D

B

H

C

Y

O,EG=EO,Y"X"0,Y"-X"0,x,=w-w'.

O, GE=180°-X"o,y"=180° —w, and therefore sin. O, GE=

sin. w.

FPE=Y,0,Y"=X"0,Y"-X"0,Y,=w-w".
PFE=0,GE.. sin. PFE sin. O, GE=sin. w.

=

sin. w

x =x′ +~, sin. (w—w')+y, sin. (w — w')

sin. w

y = y′+ *, sin. w’+y, sin, w'

y=

sin. w

are the most general formulæ for the transformation of coordinates. They are not, however, so frequently employed

as other less general formula which are deducible from them, as particular cases.

20. Case I. Let the primitive axes be rectangular and the new axes oblique.

In this case w=90° ... sin. w=sin. 90°=1;

sin. (w-w')=sin. (90°-w')=cos. ';
sin. (-)=sin. (90°-w")=cos. "

.'.x=x'+x, cos. w'+y, cos. w'
y=y'+x, sin. '+y, sin. w".

21. Case II. Let the primitive axes be oblique and the new axes rectangular.

In this case x,o, Y,=90°... w" =w' +90°

..w-w"-w-(w' + 90°) = −90° + (w−w')

and sin. (ww")=sin. (— 90° + (w-w')) =cos. cos. (w-w') (Trigon. Art. 31.)

also sin. w"-sin. (w'+90°)=cos. ∞'

22. Case III. Let the primitive axes be rectangular, and the new axes also rectangular.

sin. (w-w')=sin. (90°-')=cos. w';

sin.(w—w′′) = sin. {90°—(90°+w ́)}=sin.(—w')=—sin.w';

sin. sin. 90°=1;

sin. w"=sin. (90° +w′)=cos. w' ;

..x=x'+x, cos. ∞'—y, sin, ∞';

y=y'+x, sin. w'+y, cos. w'.

23. Case IV. Let the primitive and the new axes be parallel to each other.

In this case '=0 and "w

... sin. (w—w')=sin. w; sin. (w-w")=0; and sin. w'=0