Equation of Envelope of Family of Curves/Examples/y = 2mx + m^2

Examples of Equation of Envelope of Family of Curves

$\color {blue} {y = 2 m x + m^2} \\ \color {red} {y = -x^2}$

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:

$y = -x^2$

$\ds y$

$=$

$\ds 2 m x + m^2$

$\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

$\blacksquare$