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 . then SG : SP :: CS : CA. G 82 CONIC SECTIONS. Troduce SP to W; then since PG bisects the angle SPIV, {Prop. IX.) SG : SG :: SP : SP,... £# : SG - SG :: £P : SP - SP,but SP - SP = A A, {Prop. III.)and SG - SG = SS, .-. SG : SS :: SP : ^,or,SG : £P :: #£ : A A,or £G : SP :: (7£ : Hence also, SG : SP:: OS : CA. Prop. XII. 53. If from the foci S and S of an hyperbola S Y andS Y are drawn at right a
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 . then SG : SP :: CS : CA. G 82 CONIC SECTIONS. Troduce SP to W; then since PG bisects the angle SPIV, {Prop. IX.) SG : SG :: SP : SP,... £# : SG - SG :: £P : SP - SP,but SP - SP = A A, {Prop. III.)and SG - SG = SS, .-. SG : SS :: SP : ^,or,SG : £P :: #£ : A A,or £G : SP :: (7£ : Hence also, SG : SP:: OS : CA. Prop. XII. 53. If from the foci S and S of an hyperbola S Y andS Y are drawn at right angles to the tangent at P, then Y andI are on the circumference of the circle described on A Aas diameter, and SY . SY = PC2. Join SP, SP, and produce SY to meet SP in W ; joinCY; then since the angle SP Y = the angle WP Y, {Prop. VI.) and the angle SYP = the angle WYP, and the side PY is common to the triangles SPY, WPY, .-. the triangle SPY - JFPFin all respects, .-. SP = PJV,andSY= WY, .-. SP - SP = /STF, but £P - SP = AA, {Prop. III.) .-. SW = AA. Again, .-. SO = CS, and SY= IVY, .-. £C : CS :: £F : YW, .-. CY is parallel to SW, .-. CT : SW :: C£ :: SS, CONIC SECTIONS. 83. .-. GY = $SW = CA-so GY = CA. .. Y and Y are points on the circumference of the circledescribed upon A A as diameter. Next, let^ybe produced to meet this circle inZ, and joinZY; then since the angle Z YY is a right ZY passes through the centre C,.-. the angle 8CZ = the angle SCY,.-. SZ = 5F,.-. #r . #T = sr. &Z, = SA . SA, (Euclid, III. 36 Cor.)= CS2 - CA2, (Euclid, II. 6.)= BC\ Coe. If (7D be drawn parallel to the tangent at P meetingSPmE; then since CEPYis a parallelogram, .-. PE = CY = AC G 2 84 CONIC SECTIONS. Prop. XIII. 54. To draw a pair of tangents to an hyperbola from anexternal point 0.
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
