An Elementary Treatise On Conic Sections And Algebraic Geometry, With Numerous Examples And Hints for Their Solutions, Especially Designed for the Use of Beginners

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Putting x = aein the equation to the curve, we have or y = + - ; hence the latus rectum = — = 2a (1 — e") in the ellipse, and = 2a (e* — 1) in the hyperbola.
*201. In the ellipse the normal bisects the interior angle between the focal distances, and in the hyperbola the exterior angle; and the focal radii make equal angles with the tangent.
The equation to HP, since it passes through {x'y), (ae, 0) is -^=^y- (1), x — ae X —ae ^ ' and similarly, the equation to 8P\b ^JL^ = ^Jl— (2).
X ■{■ ae X ■
...{■ ae Hence, forming the equation to the bisector PG of the angle between (1) and (2), we have, by the rules of Art. 59; — {fC — ae) y + y'x — aey _ (a;' + ae) y — y'x — aey' .„.
{y'-+{x--aey]k ~ {y^+(a,' + ae)'}i --W.
which we might shew by reduction to be the equation to the normal at {x'y"). This may however be proved briefly as fol- lows. The denominators of (3) are evidently HP and SP, and therefore reduce, for the ellipse, to a — ea;' and a + ea;', as in Art.


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