Cartesian Equation of Conchoid of Nicomedes/Also presented as
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Cartesian Equation of Conchoid of Nicomedes: Also presented as
The directrix of $\KK$ can also be seen expressed in Cartesian coordinates in the form:
- $\paren {a - b - x} \paren {a + b - x} x^2 + \paren {a - x}^2 y^2 = 0$
Proof
| \(\ds \paren {a - b - x} \paren {a + b - x} x^2 + \paren {a - x}^2 y^2\) | \(=\) | \(\ds \paren {\paren {a - x} - b} \paren {\paren {a - x} + b} x^2 + \paren {a - x}^2 y^2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\paren {a - x}^2 - b^2} x^2 + \paren {a - x}^2 y^2\) | Difference of Two Squares | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {x - a}^2 x^2 - b^2 x^2 + \paren {x - a}^2 y^2\) | rearranging | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {x - a}^2 \paren {x^2 + y^2} - b^2 x^2\) | rearranging |
from which it follows that the given form is equivalent to:
- $\paren {x - a}^2 \paren {x^2 + y^2} = b^2 x^2$
$\blacksquare$
Sources
- Weisstein, Eric W. "Conchoid of Nicomedes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConchoidofNicomedes.html