. Algebraic geometry; a new treatise on analytical conic sections . Fig. 139. Second method. Take the centre C, and join CP. , Draw any chord parallel to CP and bisect it at V. Join CV,and draw PT parallel to CV. PT is the tangent at P, for it isparallel to CV which is the diameter conjugate to CP. 216 PROPERTIES OF THE ELLIPSE. [chap. x. 237. To draw tangents to an ellipse from an external point T. T K -?==—°\ 2\\ /^ \y l^i\ ^ ^ ( t \L^?> v_ \ ^ X First method. Flo. 140. Draw TK perpendicular to the directrix, and with centre S (thecorresponding focus),and radius e. TK, de-scribe a circle.
. Algebraic geometry; a new treatise on analytical conic sections . Fig. 139. Second method. Take the centre C, and join CP. , Draw any chord parallel to CP and bisect it at V. Join CV,and draw PT parallel to CV. PT is the tangent at P, for it isparallel to CV which is the diameter conjugate to CP. 216 PROPERTIES OF THE ELLIPSE. [chap. x. 237. To draw tangents to an ellipse from an external point T. T K -?==—°\ 2\\ /^ \y l^i\ ^ ^ ( t \L^?> v_ \ ^ X First method. Flo. 140. Draw TK perpendicular to the directrix, and with centre S (thecorresponding focus),and radius e. TK, de-scribe a circle. From T, draw tan-gents TR, TR to thiscircle. Join SR, SR andlet them meet theellipse at Q, Q respec-tively. TQ, TQ aretangents at Q and Qrespectively. The proof is similarto that in Art. 171. Second method. Draw the auxiliary Fio. 141. circle. On ST as diameter describe a circle cutting the auxiliary circle at Y and Z. Join TY, T2. TY, TZ are tangents to the AET. 239.] PROPERTIES OF THE ELLIPSE. 217 Y is a point on the auxiliary circle and l SYT is the angle in asemi-circle and therefore a right TV is a tangent. (Art. 227.)Similarly TZ also is a tangent. 238. Given a curve, which is hnovm to he an ellipse, find its centre,the positions and lengths of its principle axes, and its fad. Draw any two parallel chords and bisect them at V and VV produced meet the curve at P and P. PP is a diameter,for it bisects parallelchords. .. C the middlepoint of PP is thecentre. On PP as diameterdescribe a circlemeeting the ellipseat Q. lPQP is art. L, and PQ, PQ aresupplemental chords. .. the diametersACA, BCB parallel toJ^Q and PQ are con-jugate diameters atright angles, they are the principal ^^^ ^^^ axes. With centre B and radius equal to CA describe a circle cuttingAA at S and S. BS = CA = BS. .. S and S are the foci. 239. If PCP, DCD are two conjugate diameters of an ellipse Cp2 + CD2 = a2-)-62_ Let 9 be the eccentric angle of P;
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