Equation of Limaçon of Pascal/Cartesian Form
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Theorem
The limaçon of Pascal can be defined by the Cartesian equation:
- $\paren {x^2 + y^2 - a x}^2 = b^2 \paren {x^2 + y^2}$
Proof
\(\ds r\) | \(=\) | \(\ds b + a \cos \theta\) | Equation of Limaçon of Pascal: Polar Form | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds r^2\) | \(=\) | \(\ds r b + r a \cos \theta\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2 + y^2\) | \(=\) | \(\ds b \sqrt {x^2 + y^2} + a x\) | Conversion between Cartesian and Polar Coordinates in Plane | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^2 + y^2 - a x\) | \(=\) | \(\ds b \sqrt {x^2 + y^2}\) | rearranging | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x^2 + y^2 - a x}^2\) | \(=\) | \(\ds b^2 \paren {x^2 + y^2}\) | rearranging |
$\blacksquare$