Cartesian Equation of Conchoid of Nicomedes

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Theorem

Let $a \in \R$, $b \in \R_{>0}$ be real constants.

Let the focus point of a conchoid of Nicomedes $\KK$ be located at the origin of a Cartesian plane.

Let the directrix of $\KK$ be the straight line $x = a$.


Then $\KK$ can be expressed in Cartesian coordinates as:

$\paren {x - a}^2 \paren {x^2 + y^2} = b^2 x^2$


Proof


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Also presented as

The directrix of $\KK$ can also be seen expressed in Cartesian coordinates in the form:

$\paren {a - b - x} \paren {a + b - x} x^2 + \paren {a - x}^2 y^2 = 0$


Also see


Sources

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