A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . in CY; then since the angle SPY = the angle 7FPF, (Prop. VII. Cor.) and the angle £ZP = the angle WYP, and the side PF is common to the triangles SPY, WPY, .. the triangle SPY = the triangle WPY in all respects, . -. SP = P IF,.-. SP + ,SP = S .SP + SP = A A, (Prop. III.) Again ?.• aSTC = SC and £F = FJF, .-. SG : SC :: SY : YW, .-. CY is parallel to SW, .-. CY : SW :: GS : SS, .-. C7=i >srxr =
A geometrical treatise on conic sections, with numerous examplesFor the use of schools and students in the universitiesWith an appendix on harmonic ratio, poles and polars, and reciprocation . in CY; then since the angle SPY = the angle 7FPF, (Prop. VII. Cor.) and the angle £ZP = the angle WYP, and the side PF is common to the triangles SPY, WPY, .. the triangle SPY = the triangle WPY in all respects, . -. SP = P IF,.-. SP + ,SP = S .SP + SP = A A, (Prop. III.) Again ?.• aSTC = SC and £F = FJF, .-. SG : SC :: SY : YW, .-. CY is parallel to SW, .-. CY : SW :: GS : SS, .-. C7=i >srxr = ca. So CF = CA,. \ Fand F are points on the auxiliary circle. Next let YS he produced to meet the auxiliary circle inZ, and join Z Y; then since the angle ZYY is a right angle,. ?. Z Y passes through the centre C,. -. the angle SCZ = the angle : SZ= SY,.-. ST: SY = ST . SZ, = AS. AS, (Euclid, III. 35)= CA2 - GS*, (Euclid, II. 5)= EC2. (Prop. IV.) Cor. If CD be drawn parallel to the tangent at P, meetingSP in E ; then 46 CONIC SECTIONS. since the figure CYPE is a parallelogram,.-. PE = CY = AC. Prop. XVI. 34. To draw a pair of tangents to an ellipse from anexternal point With centre S and radius equal to A A descnoe a circle. Join OS, OS; and let SO or SO produced meet the circlein the point I. Now, if 0 be a point outside the circle MIM, it is evidentthat OS is greater than 01; and if 0 be inside the circle, since OS + OS > A A or SI, (Prop. V.) .-. 0S> 01. With centre 0 and radius OS describe another circlecutting the former in the points M and M, which it willalways do, since OS is greater than 01. CONIC SECTIONS. 47 Join SM, SM, meeting the ellipse in the points P and OP, OP; these will be the tangents SP, SP; then, since SP + SP = AA = SM,.-. SP = v SP,PO = MP, PO, each to each,and OS = OM,.. the angle OPS = the angle OPM,.-. OP is the tangent at P. (Prop. VII. Cor.)So OP is the tangent at P. Pkop. XVII. If from a point 0 a pai
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
