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Cissoid of Diocles Catacaustic

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For the parametric representation

x=(2t^2)/(1+t^2)
(1)
y=(2t^3)/(1+t^2),
(2)

the catacaustic of this curve from the radiant point (8a,0) is given by

x=-(4t^2(t^2-1))/((t^2+1)^2)
(3)
y=(8t^3)/((t^2+1)^2).
(4)

Eliminating t gives the Cartesian equation

(x^2+y^2+2x)^2-4(x^2+y^2)=0.
(5)

Therefore, since

(x^2+y^2-ax)^2-a^2(x^2+y^2)=0.
(6)

is the equation of a cardioid, the catacaustic of the cissoid of Diocles for radiant point at (8a,0) is a cardioid with a=-2.

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