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. 1896 Excerpt: ...m1m3 + m2(m1--m3) = m1m3-m22 (6). Also, (5) and (2) give 2m22 = (w + m3)2-2m1m3=m22-2m1m3, i.e. 7W22 + 2m1m3=0 (7). Solving (6) and (7), we have 2a-ft, " rt 2a-h 7713=--=--, and 7M2 =-2 x. oa oa Substituting these values in (4), we have.2a--h h 2a-h I 2-" oa a i.e. 21ak2 = 2(h-2a) so that the required locus is 21ay2=2 ...
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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. 1896 Excerpt: ...m1m3 + m2(m1--m3) = m1m3-m22 (6). Also, (5) and (2) give 2m22 = (w + m3)2-2m1m3=m22-2m1m3, i.e. 7W22 + 2m1m3=0 (7). Solving (6) and (7), we have 2a-ft, " rt 2a-h 7713=--=--, and 7M2 =-2 x. oa oa Substituting these values in (4), we have.2a--h h 2a-h I 2-" oa a i.e. 21ak2 = 2(h-2a) so that the required locus is 21ay2=2(x-2af. 238. Ex. If the normals at three points P, Q, and R meet in a point O and S be the focus, prove that SP. SQ. SR = a. SO2. As in the previous question we know that the normals at the points (am12, -2am1), (am22, -2am2) and (am32, -2am3) meet in the point (ft, k) if m1 + m2 + m3 = 0 (1), 2a-ft m2m3 + m+711-2 =-----(2), k and m1m2m3=--(3). By Art. 202 we have 8P=al + ml2) SQ-a(l + m22)f and SR = al+m32). Hence SP'S(R)' SR = (l + m12) (l + m22) (l + m32) = 1 + (mx2 + m22 + ra32) + (m22m32 + m.m + mm?) + m-fm22m. Also, from (1) and (2), we have mx2 + m22 + m32 = (m1 + m2 + ms)2-2 (?w2m3 + m3wi + wiw2) and m2%32 + mm-f + m12m22 = (m2mB + m37?i1 + ra)2-2m1m2m3 (mx + m2 + m3) = (-)', by(l)and(2). SP.SQ.SB, 07z-2a fh-2ay k2 Hence 1 = 1 + 2----+ ( +--a? a a J a _(h-a)2 + k2_S02 a2 a2 ' i.e. SP.SQ.SR = S02.a. EXAMPLES. XXX. Find the locus of a point 0 when the three normals drawn from it are such that 1. two of them make complementary angles with the axis. 2. two of them make angles with the axis the product of whose tangents is 2. 3. one bisects the angle between the other two. 4. two of them make equal angles with the given line y=mx + c. 5. the sum of the three angles made by them with the axis is constant. 6. the area of the triangle formed by their feet is constant. 7. the line joining the feet of two of them is always in a given direction. The normals at three points Pf Q, and 12 of the parabola y2--4ax meet in a point 0 who...
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