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 . tangles; and therefore the angle ZSP being half of theangle P SQ, becomes ultimately a right angle. Hence when PZ becomes the tangent at the point P,the angle ZSP is a right angle,or SZ is perpendicular to SP. Coe. 1. Conversely, if SZhe drawn at right angles to SP,meeting the directrix in Z, and PZ be joined, PZ will be thetangent at P. Con. 2. If ZP be produced to meet the other directrix onthe point Z
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 . tangles; and therefore the angle ZSP being half of theangle P SQ, becomes ultimately a right angle. Hence when PZ becomes the tangent at the point P,the angle ZSP is a right angle,or SZ is perpendicular to SP. Coe. 1. Conversely, if SZhe drawn at right angles to SP,meeting the directrix in Z, and PZ be joined, PZ will be thetangent at P. Con. 2. If ZP be produced to meet the other directrix onthe point Z and SZ be joined, then SZ is at right angles to SP. Cor. 3. The tangents at the extremities of the latus rectumor double ordinate through the focus meet the axis producedin the point X. Prop. VII. The tangent to the ellipse at any point P makes equalangles with the focal distances SP and S P. Let the tangent at P meet the directrices in Z and Z. Draw MPM at right angles to the directrices, meetingthem in if and M respectively ; join SZ, SZ ; then SP : PM :: SP : PM; and since the triangles 31 PZ, MPZ, are similar, PM : PZ :: PM : PZ .-. SP : PZ :: SP : PZ. (Ex cequaU.) CONIC Now in the triangles SPZ, SPZ, because the sides aboutthe angles SPZ, SPZ are proportional, and the angles PSZ,PS Z are equal, being right angles, and the angles SZP,SZP are each less than a right angle, .-. the triangles SPZ and SPZ are similar, (Euclid, VI. 7).-. the angle SPZ = the angle SPZ. Cor. If SP be produced to W; then the angle SPZ = the angle WPZ. Prop. VIII. The. tangents at the extremities of a focal chord intersectin the directrix. Let PSQhe a focal chord, and let the tangent at Pnieetthe directrix in Z. Join SZ; then CONIC SECTIONS. 39 the angle Z SP is a right angle, (Prop. VI.)and .-. also the angle ZSQ is a right angle, .-. ZQ is the tangent at Q ; (Prop. VI. Cor. 1)or the tangents at the extremities of a focal chord intersect inthe directrix. Prop. IX. 29. If the tangent at P
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Keywords: ., bookcentury1800, bookdeca, booksubjectconicsections, bookyear1887
