[Illustration: FIG 2.]
If, as in Fig. 3, the given point be made to coincide with the center
of the circle, the ellipse becomes a circle with diameter A B = F O.
But if the point be placed upon the circumference, as in Fig. 4, the
ellipse will reduce to the right line A B coinciding with F O.
[Illustration: FIGS 3, 4, 5, 6.]
In this case we may also apply the same process as in Fig. 1; D T
becomes a tangent at D to the circumference, and the asymptotes
coincide with the axis of the hyperbola, of which one branch reduces
to the right line A P extending from A to infinity on the left, and
the other reduces to the right line B G Q, extending from B to
infinity on the right.
If the circle be reduced to a point, as in Fig. 5, the resulting locus
is a right line perpendicular to and bisecting D O. If on the other
hand the diameter of the given circle be infinite, the circumference,
as in Fig. 6, becomes a right line perpendicular to the axis at F, and
the curve satisfies the familiar definition of the parabola, D E being
equal to E H, D P equal to P L, and so on.
In Fig. 7, as in Fig. 1, DT is tangent at T to the given circle whose
center is O, and at t to the circle about C whose diameter is AB, the
major axis. Since DTO is a right angle, T lies upon the circumference
of the circle whose center is C, and diameter DO; this circle cuts the
asymptote SCS at M and N.
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