An elementary treatise on curve tracing . b)2 = (a2- x2)(bx - a^)^. The curve is symmetrical with respect to the axis of x. Whencc = 0, y = a^lb, x = a/b, y = (), x = a, y = 0, x = b, y=:x). If y is real, we must have —a<^x? or , 0), {l-aW)y^ = i\ near ( ± a, 0), y^= ^ 2a^, near (6, co ), x — b = (a- — b^fjay. Fig. 4. The dark lines correspond to b = a, the curves markeda. and /3 correspond to 5 > and -< a. Ex. 5. (a=5y - x*)- - a^x - 2a)2(x- - a-) = 0. There can be no value of x between a and — X = d= (X, y = a, two values, cc = 2a, y — 2*a, two values, X=co , y = co^ and ?y = 0
An elementary treatise on curve tracing . b)2 = (a2- x2)(bx - a^)^. The curve is symmetrical with respect to the axis of x. Whencc = 0, y = a^lb, x = a/b, y = (), x = a, y = 0, x = b, y=:x). If y is real, we must have —a<^x? or , 0), {l-aW)y^ = i\ near ( ± a, 0), y^= ^ 2a^, near (6, co ), x — b = (a- — b^fjay. Fig. 4. The dark lines correspond to b = a, the curves markeda. and /3 correspond to 5 > and -< a. Ex. 5. (a=5y - x*)- - a^x - 2a)2(x- - a-) = 0. There can be no value of x between a and — X = d= (X, y = a, two values, cc = 2a, y — 2*a, two values, X=co , y = co^ and ?y = 0 gives no real values of x. Near (± a, a), rf=± 2a^, near (2a, 2*a), {a^n - 4 • 2ht^if - Sa^^^ ^ q^or ~ f] = (2±JS)i. Near (oo, oc ), a^y — x^= ±a^x^. / % v^. SYSTEMATIC TEACING OF CURVES 171 PLATEThere is, therefore, a multiple point (2a, 2*a) on the xiv. parabolic asymptote o?y = x^, from which the curve runs off Y\g. both sides of the asymptote. Ex. 6. (x2-4)2-y2(y + 16) = 0. The curve is symmetrical with respect to Oy; therefore,considering only positive values of x,if i/ = Oor—16, a) = 2, two values, ifa; = 0, 2/^+161/2 = 16, whence y = \-^^, _(l+^i_), _(16-Jg) nearly; if 2/ = 00 , 33= CO . Near (2,0), (4^)2-16i/2 = 0, near (2, -16), {^if-lQS = 0. If /3 be one of the values of y when x = 0,near(0,/3), -8x2-(3/32 + 32^);; = 0,hence 4cc2+17>/ = 0, 4a;2-15>? = 0,and fl32+32;; = 0 nearly. Near (oo , oo ), a?* —2/^ = 0. It may be shewn that the curve is parallel to Oy, wherey=— ^-i and x = ^/- nearly. Fig. 6. 196. In the case of equations which cannot be solvedwith respect to either of the coordinates, or which, if capableof solution, would lead to clumsy results, no order of pro-ceeding can be laid down as being generally
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