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(D) Let the point of contact on the hyperbola $xy = c^2$ be $(ct, c/t)$.The equation of the tangent at $(ct, c/t)$ is given by $x/t + yt = 2c$,or $x +

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$(x^2 + y^2)^2 = 4c^2 xy$

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(D) Let the point of contact on the hyperbola $xy = c^2$ be $(ct, c/t)$.The equation of the tangent at $(ct, c/t)$ is given by $x/t + yt = 2c$,or $x + t^2y = 2ct$.The slope of this tangent is $m = -1/t^2$.The line passing through the origin $(0,0)$ and perpendicular to the tangent has slope $m' = t^2$.The equation of this perpendicular line is $y = t^2x$.Substituting $t^2 = y/x$ into the tangent equation $x + t^2y = 2ct$:$x + (y/x)y = 2c\sqrt{y/x} \implies x + y^2/x = 2c\sqrt{y/x}$.Multiplying by $x$,we get $x^2 + y^2 = 2c\sqrt{xy}$.Squaring both sides,we get $(x^2 + y^2)^2 = 4c^2xy$.

The locus of the foot of the perpendicular from the centre of the hyperbola $xy = c^2$ on a variable tangent is :

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