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Question

Show that all curves for which the slope at any point (x, y) on it is x2+y22xy are rectangular hyperbola.

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Solution

We have,
dydx=x2+y22xyLet y=vxdydx=v+xdvdx v+xdvdx=x2+v2x22vx2xdvdx=1+v22v-vx dvdx=1+v2-2v22vxdvdx=1-v22v2v1-v2dv=1xdx

Integrating both sides, we get2v1-v2dy=1xdx-log 1-v2=log x-log Clog 1-v2C=-log x1-v2=Cx1-yx2=Cxx2-y2x2=Cxx2-y2=Cx
Thus, x2-y2=Cx is the equation of the rectangular hyperbola.

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