The semi-conjugate axis is a mean
proportional between D A and AO; now drawing TM and TN, it is seen
that Tt is that mean proportional; and a circle described about C with
that radius will be tangent to TO. DT, then, is the radius of the
circle to be described about the focus of the conjugate hyperbola for
its construction according to the enunciation first given: and we
observe that DT and TO are supplementary chords in the circle about C
through D and O. The conjugate foci must therefore lie upon this
circumference, at D' and O'; and since D'O' is perpendicular to DO,
D'T will be perpendicular and T'O' will be parallel to SCS.
[Illustration: FIG 7.]
Now as TO increases, T'O' will diminish, until, when TO equals DO,
T'O' will vanish and with it Ct'; and at this crisis, the case is the
same as in Fig. 4; but the conjugate hyperbola logically reduces to
_two_ right lines, extending from C to infinity on the right and left.
As indeed it should from the familiar construction, since the
distances from D' and O' to any point on the horizontal axis being
equal, their difference is constant and equal to zero.
It appears, then, that a conic section may be defined as the locus of
a point which is equally distant from a given point and from the
circumference of a given circle. Boscovich defines it as the locus of
a point so moving that its distances from a given point and from a
given right line shall have a constant ratio.
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