Condition for Conchoid of Nicomedes to have Cusp
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Theorem
Let $\KK$ be a conchoid of Nicomedes such that:
- the perpendicular distance from the directrix $\CC$ of $\KK$ to the focus point $P$ of $\KK$ is $a$
- the constant distance from $\KK$ to $\CC$ measured along a straight line $\LL$ through $P$ is $b$.
Then:
- $\KK$ has a cusp at $P$
- $a = b$
Proof
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Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): conchoid
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): conchoid