Cartesian Co-ordinates. 39

29. If the equation of the line AB be given in the form

BJ

M

x cos a + y cos /5 - p, we find the length of the perpendicular
from the point M9 as follows :—
Let OP = #', PM = y') and MR the perpendicular from M
upon AB = _#/. Then the projection of 02t on OQ is equal to
the projection of the contour OPMR on OQ,. Hence,
jp = xr cos a + y1 cos ft +p', .'. -pf - xf cos a + y1 cos /3 -^,
.•. y = - the power of the point Jf. (82)
"We suppose that p' is suhject to the same rule of signs as^>;
p is always +, and the points for which p is positiv.e are on the
same side of the line as the origin of co-ordinates.
Cor.—The power of any point on a line with respect to the
line is zero ; and, conversely, if the power of a point with respect
to a line be zero, the point must be on the line.
30. If S = Ax + By + C = 0, S' = A'x + B'y 4- Cf = 0, le
the equations of any two lines, and I, m any two multiples (includ-
ing unity}) either positive or negative, then
IS + mSf=Q (83)
is the equation of some line passing through the intersection of the
lines S and S'.
For, since S and S' are of the first degree with respect to
x and y, IS + mS' = 0 will also be of the first degree, and there-
fore will be the equation of some line. Again, if P be the point
of intersection of S and S', the powers of P (§ 29, Cor.} with