. Algebraic geometry; a new treatise on analytical conic sections . PQ. .*. CV is a diameter bisecting chords parallel to PQ or CV,and CV „ „ „ PQ or CV. .. CV, CV are conjugate diameters, and are parallel to thesupplemental chords PQ, PQ. 200 THE ELLIPSE. [chap. x. 213. If an ellipse has Us centre at the origin, its equation containsno terms of the first degree. If possible let ax^ + 2hxy + by^ + 2gx+2fy + c = 0 be the equationof the ellipse. Let {x, y) be any point on the curve. Then since the origin isat the centre, {-x, -y) is also on the curve. .. ax^ + 2hxy + by^ + 2gx + 2fy + c = 0and a
. Algebraic geometry; a new treatise on analytical conic sections . PQ. .*. CV is a diameter bisecting chords parallel to PQ or CV,and CV „ „ „ PQ or CV. .. CV, CV are conjugate diameters, and are parallel to thesupplemental chords PQ, PQ. 200 THE ELLIPSE. [chap. x. 213. If an ellipse has Us centre at the origin, its equation containsno terms of the first degree. If possible let ax^ + 2hxy + by^ + 2gx+2fy + c = 0 be the equationof the ellipse. Let {x, y) be any point on the curve. Then since the origin isat the centre, {-x, -y) is also on the curve. .. ax^ + 2hxy + by^ + 2gx + 2fy + c = 0and ax^ + 2ki;y + by^-2gx-2fy + c = 0;.. by subtraction, igx + ify^O, or gx+fy = (x, y) is any point on the curve,.. gx +fy = 0 for every point on the we must have /= 0 and g = 0. Hence the equation of an ellipse referred to any axes throughthe centre is of the form ax^ + 2hxy + by^ + c = more useful form is ax^ + 2hxy + by^=l. *214. To fmd the equation of an ellipse referred to two conjugatediameters as axes of co-ordinates. (Oblique.) C-x^y/. Fio. 130. The origin being at the centre, if (x, y) lies on the curve,{-X, -y) also lies on the curve ; .. its equation can contain no terms of the first degree. ART. 216.] CONJUGATE DIAMETEES AS AXES. 201 We may therefore take ki? + 1Hxy+By^=\ . (1) to be the equation of the ellipse. Again, if P is the point {x, y) and the chord PVP is parallel toCB the axis of y, PV = PV, or the co-ordinates of P, a point onthe curve, are {x, - y); :. Aa;2 - 2 Hxy + By^ = 1 by substitution; .. from (1) by subtraction, H = 0; .. the equation of the curve reduces toAa;2 + By2=l. Again, let a, V be the lengths of the given conjugate semi-diameters. When y = 0, x = a; .. Aa2=l and fK = -^-„ x = 0, y = h; :. B62=l and 8 = ^. •• ~^^Vi~^^^ ^^^ required equation. *215. We see that the equation of an ellipse referred to a pairof conjugate diameters is of the same form as when referred to itsprincipal axes. -7j + ^ = 1 IS a tangent to t
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