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

EXAMPLES.

117

i.e. by
X2 (1 - cos2 o) + X ( + b - 2h cos o) + ab - 2 = 0,
a+ b -2h costo ab - h2
i.e. by X — X I + =0.. (4).
sin2 o sin" o..
In a similar manner the values of X which make the
right-hand side of (3) a perfect square are given by the
equation
2 a + b'- 2h' cos o' a'b' - h'2
X2 +X +. =...... (5).
sin2 e, sin2 to
Since the values of X given by equation (4) are the same
as the values of X given by (5), the two equations (4) and
(5) must be the same.
Hence we have
a+b-2hcos o a' + b' - 2h' cos w'
sin2 c sin2 w'
ab - h2 a'b' - h2
sin2 wo sin2 ac'
EXAMPLES. XVI.
1. The equation to a straight line referred to axes inclined at 30~
to one another is y=2x +1. Find its equation referred to axes
inclined at 45~, the origin and axis of x being unchanged.
2. Transform the equation 2x2 + 3 /3xy + 3y2= 2 from axes
inclined at 30~ to rectangular axes, the axis of x remaining
unchanged.
3. Transform the equation x2+ xy +y2= 8 from axes inclined at
60~ to axes bisecting the angles between the original axes.
4. Transform the equation y2 + 4y cot a - 4x= 0 from rectangular
axes to oblique axes meeting at an angle a, the axis of x being kept
the same.
5. If x and y be the coordinates of a point referred to a system of
oblique axes, and x' and y' be its coordinates referred to another
system of oblique axes with the same origin, and if the formule of
transformation be
x= mx' +ny' and y = m'x' + n'y',
m2S + n"2 - 1 rmm'
prove that -- 1 n
it2+- n2_1 - n'