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Question

If 1x4+1y4=k(x2y2), prove that dydx=x1y4y1x4

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Solution

Given,
1x4+1y4=k(x2y2)
Differentiate on both sides w.r.t k
4x321x44y321y4(dydx)=k(2x2ydydx)

2kx+4x321x4=(2ky4y321y4)dydx

kx+x31x4=(kyy31y4)dydx

x1x4+x1x4x2y2+x31x4=(y1y4+y1y4x2y2+y31y4)dydx

xx5+x(1y4)(1x4)+x5x3y21x4=(y(1y4)(1x4)+yy5x2y3+y51y4)dydx

x(1(1y4)(1x4)x2y2)1x4=y((1y4)(1x4)+1x2y2)1y4dydx

dydx=x1y4y1x4

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