This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1840 Excerpt: ...130. The locus of the feet of the perpendiculars dropped from the foci upon the tangent to an ellipse, is the circumference of the circle whose diameter is the axis major. For joining CZ (fig. 42), since SH is bisected in C, and HK in Z, CZ is parallel to SK and equal to 1 SK = AC. Also drawing SY, CQ, parallel to HZ, ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1840 Excerpt: ...130. The locus of the feet of the perpendiculars dropped from the foci upon the tangent to an ellipse, is the circumference of the circle whose diameter is the axis major. For joining CZ (fig. 42), since SH is bisected in C, and HK in Z, CZ is parallel to SK and equal to 1 SK = AC. Also drawing SY, CQ, parallel to HZ, CQ bisects YZ perpendicularly, and therefore CY = CZ--AC. 131. Since C is the center of the circle which is the locus of F and Z, and SYZ is a right angle, if YS and ZC be produced to meet in H', this will be a point in the circle; and from the equal triangles CZH, CHS, SH-HZ;.: SYHZ = SYx SH' = AS x A'S = BC (Art. 109). HZ SY SY SP Also since-----, or _---(Art. 127); multiplying this equation by the preceding, we get SIP SY = BC--, or, if SF be denoted by p and SP by r, we have between the radius vector of any point and the perpendicular on the tangent at that point from the focus, the relation p2 = 62. 132. Draw CE parallel to PY, then PC is a parallelogram, .-. PE=CZ = AC 133. Draw HI parallel to YZ, and let I SPY = a; SI r sin a--(2a--r) sin a a result which is sometimes of use. 134. If from G the foot of the normal at P (fig. 41), GL be drawn perpendicular to either focal distance, then PL = the latus rectum.., HL HN For, by similar triangles, -----;.-. HL = HN.- -= e(ae-a) (Art. 127); HP.-. PL = a--ex-(ae2-ex) = a (l-e2) = half the latus rectum. Also, drawing IjfZ perpendicular to HP to meet the tangent at P in Z, and ZX perpendicular to CA produced, we get from similar triangles HX _GL HZ PL HP g(l-e) HZGH' HPGL' HG e" Therefore Z is a point in the directrix (Art. 110). Hence if ZP7 be joined, it is a tangent at P'. Hence the tangents at the extremities of any focal chord intersect in the directrix; and the line joining their in...
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