Equation of Envelope of Family of Curves/Examples/y = 2mx + m^2
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Examples of Equation of Envelope of Family of Curves
Consider the family of curves $\FF$ embedded in the Cartesian plane defined by the equation $E$:
- $E: \quad y = 2 m x + m^2$
where $m$ is the parameter of $\FF$.
The envelope of $\FF$ is the parabola whose equation is:
- $y = -x^2$
Proof
\(\ds y\) | \(=\) | \(\ds 2 m x + m^2\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\partial y} {\partial m}\) | \(=\) | \(\ds 2 x + 2 m\) |
Setting $2 x + 2 m = 0$:
\(\ds 2 x + 2 m\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds m\) | \(=\) | \(\ds -x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds 2 \paren {-x} x + \paren {-x}^2\) | substiting for $m$ in $E$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -x^2\) | simplifying |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): envelope: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): envelope: 1.