By William Henry Besant
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Within the sequence of volumes which jointly will represent the instruction manual of Differential Geometry a slightly entire survey of the sector of differential geometry is given. different chapters will either care for the fundamental fabric of differential geometry and with learn effects (old and recent). All chapters are written by way of specialists within the quarter and comprise a wide bibliography.
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Extra info for Conic Sections, Treated Geometrically
If the tangent at P meet the directrix in F and the axis in T , the angles KSF , F T S are equal. EXAMPLES. 18 11. P SP is a focal chord, P N , P N are the ordinates, and P K, P K perpendiculars on the directrix; if KN , K N meet in L, the triangle LN N is isosceles. 12. The focal distance of a point on a conic is equal to the length of the ordinate produced to meet the tangent at the end of the latus rectum. 13. The normal at any point bears to the semi-latus rectum the ratio of the focal distance of the point to the distance of the focus from the tangent.
102. If a circle cut a parabola in four points, two on one side of the axis, and two on the other, the sum of the ordinates of the first two is equal to the sum of the ordinates of the other two points. Extend this theorem to the case in which three of the points are on one side of the axis and one on the other. EXAMPLES. 50 103. The tangents at P and Q meet in T , and T L is the perpendicular from T on the axis; prove that if P N , QM be the ordinates of P and Q, P N . QM = 4AS . AL. 104. The tangents at P and Q meet in T , and the lines T A, P A, QA, meet the directrix in t, p, and q: prove that tp = tq.
The locus of the vertices of all parabolas, which have a common focus and a common tangent, is a circle. 12. Having given the focus, the length of the latus rectum, and a tangent, construct the parabola. 13. If P SP be a focal chord, and P N , P N the ordinates, shew that AN . AN = AS 2 . Shew also that the latus rectum is a mean proportional between the double ordinates. 14. The locus of the middle points of the focal chords of a parabola is another parabola. 15. Shew that in general two parabolas can be drawn having a given straight line for directrix, and passing through two given points on the same side of the line.
Conic Sections, Treated Geometrically by William Henry Besant