. Applied calculus; principles and applications . + y^ = Saxy. ^ Ans. x -\-y = Za, x=y. 6. x + y = 2e=-y,Sit (1, 1). Ans. 3y = x-\-2, Sx + y = 4. 7. ix/ar + (y/br=^2,at(a,b). Ans. x/a + 2//& = 2, ax — by = a^ — ¥. 108 DIFFERENTIAL CALCULUS 8. Show that the sum of the intercepts of the tangent to the para-bola X* + 2/^ = a^, is equal to a. 9. Show that the area of the triangle intercepted from the co-ordinate axes by the tangent to the hyperbola, 2 xy = a?, is equal to o?. 10. Show that the part of the tangent to the hypocycloid x^ -\-y^ = a%intercepted between the axes, is equal to a. 11. Find
. Applied calculus; principles and applications . + y^ = Saxy. ^ Ans. x -\-y = Za, x=y. 6. x + y = 2e=-y,Sit (1, 1). Ans. 3y = x-\-2, Sx + y = 4. 7. ix/ar + (y/br=^2,at(a,b). Ans. x/a + 2//& = 2, ax — by = a^ — ¥. 108 DIFFERENTIAL CALCULUS 8. Show that the sum of the intercepts of the tangent to the para-bola X* + 2/^ = a^, is equal to a. 9. Show that the area of the triangle intercepted from the co-ordinate axes by the tangent to the hyperbola, 2 xy = a?, is equal to o?. 10. Show that the part of the tangent to the hypocycloid x^ -\-y^ = a%intercepted between the axes, is equal to a. 11. Find the slope of the logarithmic curve x = logb y. The slopevaries as what? What is the slope of the curve x = logy? 12. Find the normal, subnormal, tangent, and subtangent of thecatenary y = a/2 {e^^ + e-^/*). Vy^ - a2 V 2/2 - a213. At what angles does the line Sy — 2x — 8 = 0 cut the parabola= 8 X? Ans. arc tan ; arc tan 77. Polar Subtangent, Subnormal, Tangent, Normal. — Let arc mP = s, and arc PQ = As; then z POQ = Ad,. T circular arc PM = pA0, and MQ =- Ap. The chords PMand PQ, the tangents RPU and TPZ, are drawn; and ZRis drawn perpendicular to PR, Z being any point on thetangent PZ. When As = 0, the hmiting positions of the secants PMand PQ are the tangents RPR and TPZ, respectively; hence,li (z PMQ) = z RPK = 7r/2 = z PRZ,Itiz OQP) = z OPT = tA = Z RZP,and It z MPQ = zRPZ. POLAR SUBTANGENT 109 Now in a problem of limits the chord of an infinitesimal arccan be substituted for the arc, since the limit of their ratio isunity (Art. 22 and Cor., Art. 46); so Ap^ MQ ^ sin MPQ^ As chord PQ sinPMQ , . , nA(9 , chord MP , sinMQP Agam, It-r— = It -j T ^^ = tt-—r^T.^^; ^ As chord PQ smPMQ From (1) and (2), it follows that, if PZ is taken as ds,ds = PZ, dp = HZ, and pdd = HP. Drawing OT perpendicular to OP, and PA and ON perpen-dicular to the tangent TP, the length PT is the polar tan-gent; PA, the polar normal; OA, the polar subnormal; andOT, the polar suhtan
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