The envelope of a family of curves $F(x, y, m) = 0$ is found by solving the system of equations $F(x, y, m) = 0$ and $\frac{\partial F}{\partial m} = 0$. The given equation is $y - mx - \frac{a} {m} = 0$. Differentiating with respect to the parameter $m$: $-x + \frac{a}{m^2} = 0$. This gives $m^2 = \frac{a}{x}$. Substituting this back into the original equation: $y = (\pm \sqrt{\frac{a}{x}}) x + a (\pm \sqrt{\frac{x}{a}}) $. This simplifies to $y = \pm 2\sqrt{ax}$. Squaring both sides yields $y^2 = 4ax$.
Envelope of the curve $y = mx + \frac{a}{m} $, where m is a parameter is -
Asked in Asst. Statistical Officer (Agri. Deptt.) 2024 · Topic: Wave Motion (General Wave Equation (Harmonic Waves))
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