The elements of coordinate geometry, by S. L. Loney, M.A.

114

COORDINATE GEOMETRY.

*133. Particular cases of the preceding article.
(1) Suppose we wish to transfer our axes from a
rectangular pair to one inclined at an angle o'. In this
case o is 90~, and the formulae of the preceding article
become
x= x' cos 0 + y' cos (a' + 0),
and y = x' sin 0 + y' sin (o' + 0).
(2) Suppose the transference is to be from oblique
axes, inclined at o, to rectangular axes. In this case w' is
90~, and our formulae become
x sin o = x sin (o - ) - y' cos (o - 0),
and y sin = x' sin 0 + y' cos 0.
These particular formulae may easily be proved independently, by drawing the corresponding figures.
x2 y2
Ex. Transform the equation -2 - =1 from rectangular axes to
axes inclined at an angle 2a, the new axis of x being inclined at an angle
- a to the old axes and sin a being equal to -— b
Here 0= -a and w'=2a, so that the formulae of transformation
(1) become
x = (x' + y') cos a and y = (y' - x') sin a.
b a
Since sin a=,, we have cos a= _, and hence the
^a2+ b2 a2 + b2
given equation becomes
(x'+y')2 (y'-')2
a2 — b2 a2+b2 -
i.e. x'y'= (a2+ b2).
*134. The degree of an equation is unchanged by any
transformation of coordinates.
For the most general form of transformation is found
by combining together Arts. 128 and 132. Hence the
most general formula of transformation are
sin (w a- 0),sin ( - ' - 0)
x=h+ x' - i +y -Ysin o sin o)
sin 0 sin (w'+ 0)
and y = k+x -. - Fy
sin ( sin tco