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

CHAPTER XV.

GENERAL EQUATION OF THE SECOND DEGREE.
TRACING OF CURVES.
348. Particular cases of Conic Sections. The
general definition of a Conic Section in Art. 196 was that
it is the locus of a point P which moves so that its distance
from a given point S is in a constant ratio to its perpendicular distance PM from a given straight line ZK.
When S does not lie on the straight line ZK, we have
found that the locus is an ellipse, a parabola, or a hyperbola
according as the eccentricity e is < = or > 1.
The Circle is a sub-case of the Ellipse. For the
equation of Art. 139 is the same as the equation (6) of
Art. 247 when b2=a2, i.e. when e=0. In this case
CS=O, and SZ= — ae =oo. The Circle is therefore a
e
Conic Section, whose eccentricity is zero, and whose directrix is at an infinite distance.
Next, let S lie on the straight line ZIC, so that S and Z
coincide.
In this case, since K
SP =e. PM, U
we have M P
sin PSM = P.M- I '
If e> 1, then P lies on one or
other of the two straight lines SU K U
and SU' inclined to KK' at an angle
sin-' I.