Framed Print of Plane and solid analytic geometry

Plane and solid analytic geometry. xes of an ellipse wemay mean either the transverse and conjugate axes, indefinitestraight lines, or the major and minor axes, the segments ofthese lines intercepted by the ellipse ; cf. the dual definition, Ch. VII, § 1. By a diameter of an ellipse we may mean, also, one of twothings, either an indejinite straight line through the center of theellipse, or the segment of this line intercepted by the ellipse; andwe agree to adopt this dual definition. The length of the seg-ment is called the le7igth of the diameter; its end points, theextremities of the diame. Plane and solid analytic geometry. xes of an ellipse wemay mean either the transverse and conjugate axes, indefinitestraight lines, or the major and minor axes, the segments ofthese lines intercepted by the ellipse ; cf. the dual definition, Ch. VII, § 1. By a diameter of an ellipse we may mean, also, one of twothings, either an indejinite straight line through the center of theellipse, or the segment of this line intercepted by the ellipse; andwe agree to adopt this dual definition. The length of the seg-ment is called the le7igth of the diameter; its end points, theextremities of the diameter. Problem. What is the locus of the mid-points of a set ofparallel chords of an ellipse ? i. In the special case of a circle, the locus is a diameter, con-sidered as a line-segment. This is true, also, for the generalellipse. For, if the chords are parallel to an axis of the ellipse, the theorem is geometricallyobvious; if they are oblique to theaxes, as is generally the case, weresort to an analytical proof.Let the ellipse be.. Fig. 1 (1) b a2 and let X (=?t: 0) be the slope of the chords. Consider a variablechord of slope X moving always parallel to itself. Its mo- 288 DIAMETERS. POLES AND POLARS 289 tion we express analytically by taking jS, its intercept on theaxis of y, as auxiliary variable. The equation of the chord is, then. (2) y=Xx-{-l3, where is constant and (3 is variable. The work now proceeds according to the method ofCh. XIII, § 5. If in (1) we set for y its value as given by(2), we obtain the equation or (a2X2 -f 62)aj2 + 2 a^XfSx -- af3 - b^) = 0, whose roots are the abscissae of the two points of intersectionof the line (2) with the ellipse. Half the sum of these rootsis Xy the abscissa of the mid-point, P, of the chord. Hence, by the formula, Ch. XIII, § 5, (1), for the sum of the roots ofa quadratic equation, (3). X= ^^M_. Since, moreover, P: (X, T) lies on the chord (2), we have (4) -^ T=XX--l3. It remains to eliminate (3 from (3) and (4). Subs. © Reading Room 2020

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