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 . r-sect at right angles in the directrix. Let PSQ be a focal chord, and let the tangent at Pmeetthe directrix in Z. Join SZ; then the angle ZSP is a right angle, (Proj). IV.)and .. also the angle ZSQ is a right angle,.•. Z Q is the tangent at Q, {Prop. IV. Cor.) or the tangents at the extremities of the focal chord PSQintersect in the directrix. Again, draw PM, QM at right angles to the directrix ;then si
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 . r-sect at right angles in the directrix. Let PSQ be a focal chord, and let the tangent at Pmeetthe directrix in Z. Join SZ; then the angle ZSP is a right angle, (Proj). IV.)and .. also the angle ZSQ is a right angle,.•. Z Q is the tangent at Q, {Prop. IV. Cor.) or the tangents at the extremities of the focal chord PSQintersect in the directrix. Again, draw PM, QM at right angles to the directrix ;then since MP, PZ = SP, PZ, each to each, and the angle MPZ = the angle 8PZ, . ?. the angle MZP — the angle 8ZP, .-. the angle SZPis half of the angle SZM. So the angle SZQ is half of the angle SZM, .-. the angle PZQ is half of the two SZM and SZM. But the angles SZM and SZM = two right angles, .. the angle PZQ is a right angle, or the tangents at the extremities of a focal chord intersectat right angles in the directrix. 10 CONIC SECTIONS. Prop. VII. 9. If the tangent at any point P of a parabola meet theaxis produced in the point T, and PN be the ordinate of thepoint P, then NT = Join SP, and draw PAf at right angles to the directrix;then .-. the angle SPT = the angle MPT = the angle STP, .-. ST = SP = PM = XX,.-. ST = A^But^l>S = AX,.-. the remainder ^4 2= the remainder AN,. = 2 AX. Def. The line NT is called the Subtangent. 10. Dee. The line PG, drawn at right angles to PT, iscalled the Normal at the point P, and NG the Subnormal. Prop. VIII. If the normal at the point P of a parabola meet the axisin the point G, then NG = 2AS. CONIC SECTIONS. H Since the angle SPG = the complement of the angle SPT,and the angle SGP - the complement of the angle STP,and also the angle SPT = the angle STP, (Prop. VII.).-. the angle SPG = the angle SGP,.-. SG= SP=PM=XN,.-. SG = away the common part SN, the remainder JV6r = SX = 2AS. Prop. IX. 11. If PJV be an
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
