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

220 COORDINATE GEOMETRY.
The equation to the tangent at (x', y') is then obtained
by putting x" = x' and y" = y', and is
y - y, (X - XI),
'.e. 7 < +< =1.........(5).
d'a ' + V Py b
This is the required equation.
[In the foregoing we have assumed that (x', y') lies on the portion
PAQ (Fig., Art. 243). If it lie on either of the other portions the
proper signs must be affixed to the radicals, as in Art. 240.]
Ex. To find the condition that the straight line,+ Y=1 may be a
f g
tangent.
This line will be the same as (5), if
f =,/ax and g=Jby',
go that v/ —i, and
f an
Hence - + =1.
a b
This is the required condition; also, since a'=f2 and y'a b
the point of contact of the given line is (f, ib:).
Similarly, the straight line lx+mzy= n will touch the parabola if
n n
a-+ =1.
al bm
243. To find the focus of the parabola
A+A~ =1.
Let S be the focus, 0 the origin, and P and Q the
points of contact of the parabola with the axes.
Since, by Art. 230, the triangles OSP and QSO are
similar, the angle SOP= angle SQO.
Hence if we describe a circle through 0, Q, and S, then,
by Euc. III. 32, OP is the tangent to it at O.