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Question

If x=etsint,y=etcost are parametric equations, then d2ydx2 at (1,1) is equal to :

A
12
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B
14
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C
0
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D
12
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Solution

The correct option is A 12
x=etsint, .... (i) and y=etcost .... (ii) are the parametric equations
At point (1,1), we get
1=etsint and 1=etcost
tant=1t=π4
Now, dydt=et(costsint)
and dxdt=et(sint+cost)

Also, dydx=dydtdxdt=costsintsint+cost
d2ydx2=ddt(dydx)dtdx=ddt(costsintcost+sint)dtdx

=(cost+sint)(sintcost)(costsint)(sin+cost)(cost+sint)2dtdx

=(cost+sint)2(costsint)2(cost+sint)2dtdx

=2(cost+sint)2.1et(sint+cost)

=2(etcost+etsint).1(cost+sint)2

=2x+y.1(cost+sint)2

d2ydx2(1,1)=21+1.1(cosπ4+sinπ4)2=12

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