. Plane and solid analytic geometry . redcoordinates: n . , , _ 0^1 -f fxxo ,,f_yi + f^y2. Vi • Xi — —— 2/1 — ——■ , 1 H- /X 1 -f /A o- ^ / _ ^1 — /^^2 „/_fcii^. ^2 ^2 —^ J 2/2 — —:; l—fX 1—fX I DIAMETERS. POLES AND POLARS 311 EXERCISES 1. Four points Pi, P2, Qi, Q2 on the axis of x have, respec-tively, the abscissas 3, 8, 5, — 7. Show that Qi, Q2 divide P1P2harmonically, and find the common ratio /a of internal and ex-ternal division. Find, also, the value of the ratio, fx, for thedivision by Pj and P2 of the segment Q1Q2 2. Find the point on the axis of x which, with the point(—1, 0), divides
. Plane and solid analytic geometry . redcoordinates: n . , , _ 0^1 -f fxxo ,,f_yi + f^y2. Vi • Xi — —— 2/1 — ——■ , 1 H- /X 1 -f /A o- ^ / _ ^1 — /^^2 „/_fcii^. ^2 ^2 —^ J 2/2 — —:; l—fX 1—fX I DIAMETERS. POLES AND POLARS 311 EXERCISES 1. Four points Pi, P2, Qi, Q2 on the axis of x have, respec-tively, the abscissas 3, 8, 5, — 7. Show that Qi, Q2 divide P1P2harmonically, and find the common ratio /a of internal and ex-ternal division. Find, also, the value of the ratio, fx, for thedivision by Pj and P2 of the segment Q1Q2 2. Find the point on the axis of x which, with the point(—1, 0), divides harmonically the segment of the axis joiningthe points (- 8, 0), (3, 0). 3. Exercise 1, for the four points Pi, P2, Qi, Q2 with therespective coordinates (2, 3), (—1, 9), (1, 5), (5, —3). .4. Find the point which, with the point (2, 1), divides har-monically the line-segment joining the points (5, —2), (1, 2). 9. Polar of a Point. Consider the following locus ellipse (1) x^ +^=1 62. Fig. 20 and the point Pi: (xi, y^) are given. A line L is drawn through Pi meeting the ellipse in Q^ and Q2, and on L the point P: (X, Y) is marked which, with Pi, divides Q1Q2 harmonically. What is the locus of P, as L revolves about Pi ? Since Pi, P divide Q1Q2 harmonically, Q^, Q2 divide PiPharmonically. Hence the coordinates (x/, ?//), (x2j 2/2O of Qi,Q2 are: Qi: Q2: r _Xi-{-jxX iCi = Xo 2/i + /xF 1+/X 1 +/X , _ Xi-,xX , _ Vi- txT—^ 5 2/2 -—j i — fX 1 — [X. As L rotates, the ratio /x varies ; it is, then, an auxiliaryvariable expressing analytically the rotation of L. 312 ANALYTIC GEOMETRY The coordinates of Qi and Q2 satisfy (1). Substituting themin turn in (1) and clearing each of the resulting equations offractions, we have To eliminate fx, we subtract the second equation from the first,thus getting 4 fxb~XiX + 4 /Aa2?/i r = 4 ^a^b^ or, finally, ^ + ^ = 1. a^ b^ The locus of P is, therefore, a straight line, or a portion ofa straigh
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