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Question Number 93483 by Rio Michael last updated on 13/May/20

$$\mathrm{prove}\mathrm{that}\mathrm{the}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{normal}\mathrm{to}\mathrm{the}\mathrm{rectangular}$$$$\mathrm{hyperbola}xy=c^2\mathrm{at}\mathrm{the}\mathrm{point}P(ct,c/t)\mathrm{is}{t}^{3}x-ty=c(t^4-1).$$$$\mathrm{the}\mathrm{normal}\mathrm{to}P\mathrm{on}\mathrm{the}\mathrm{hyperbola}\mathrm{meets}\mathrm{the}\mathrm{x}-\mathrm{axis}\mathrm{at}Q\mathrm{and}\mathrm{the}$$$$\mathrm{tangent}\mathrm{to}P\mathrm{meets}\mathrm{the}\mathrm{yaxis}\mathrm{at}R.\mathrm{show}\mathrm{that}$$$$\mathrm{the}\mathrm{locus}\mathrm{of}\mathrm{the}\mathrm{midpoint}\mathrm{oc}QR,\mathrm{as}P\mathrm{varies}\mathrm{is}2c^{2}xy+y^4=c^4.$$

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