VI.—Note on Confocal Conic Sections
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by
H. F. Talbot
1865 Volume 24, Issue 01, p53-57
Abstract
A short paper of mine on Fagnani's theorem, and on Confocal Conic Sections, was inserted in the twenty-third volume of the Transactions of the Royal Society. Some of the conclusions of that paper can, however, be obtained more simply, as I will now proceed to show.I will, in the first place, resume the problem—"To find the intersection of a confocal ellipse and hyperbola."Since the curves have the same foci, and therefore the same centre, let the distance between the centre and focus be called<jats:italic>unity</jats:italic>, since it is the same for both curves. Let<jats:italic>a, b</jats:italic>, be the axes of the ellipse, A, B, those of the hyperbola. Then we have 1 =<jats:italic>a</jats:italic><jats:sup>2</jats:sup>−<jats:italic>b</jats:italic><jats:sup>2</jats:sup>= A<jats:sup>2</jats:sup>+ B<jats:sup>2</jats:sup>, which equation expresses the condition of confocality.
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