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If, x=a secα ,y=b tanα then find d²y/dx²

James Pradhan 1 month, 1 week ago

Solution :- x= a sec alpha , d(x)/d(alpha)=a sec alpha.tan alpha....(i) And y=b tan alpha, then, d(y)/d(alpha)= b sec²alpha ....(ii). Now we have to find, dy/dx={ dy/d(alpha)}/{dx/d(alpha)}=( bsec² alpha) /(asec alpha.tan alpha)=b/a sec alpha/tan alpha = b/a .1/cos alpha/sin alpha/ cos alpha = b/a .cosec alpha. Then d²/dx²= d(b/a.cosec alpha)/dx=b/a.(-cosec alpha.cot alpha).d(alpha)/ dx =-b/a.cosec alpha.cot alpha×1/dx/d(alpha)=-b/a.cosec alpha.cot alpha×1/a sec alpha.tan alpha=-b/a². 1/sin alpha.cot alpha×cos alpha×cot alpha=-b/a² cot³alpha (Ans.)

Pallavi Prasad 1 month, 2 weeks ago

x=a secα dx = a secα tanα dα......(i) And y=b tanα dy = b sec²α dα.......(ii) Dividing both equations dy/dx= a/b cosecα Again differentiate wrt to x d²y/dx²= a/b (-cotα cosecα)×dα/dx = a/b(-cotα cosecα)×1/a secα tanα =-cot³α/b

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