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Question

Find dydx, if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

x=a(cos t+log tant2), y=a sin t.

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Solution

Given, x=a(cos t+log tant2)

Differentiating w.r.t. t, we get

dxdt=a{sin t+1tant2.sec2t2.12} [ ddx(log|x|)=1x] =a{sin t+12sint2 cost2} =a{sin t+1sin t}=a{1sin2tsin t} ( sin t=2sin t2 cost2) dxdt=a cos2tsin t ( 1sin2t=cos2t)........(i)Also, y=a sin t dydt=a cos t .....(ii)From Eqs. (i) and (ii), dydx=dydtdxdt dydx=a cos ta cos2tsin t=tan t


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