I have an equation for an Arrhenius-like exponential curve:
$y = t\exp(-1000t/x)$
Where t is some scaling parameter. If I allow $t$ to vary from 1 to 20 in steps of 0.1, I obtain the family of curves given in the image.
The black dots in the image indicate intersection points between two neighbouring curves. I would now like to find a mathematical form of the envelope of this set of curves, but I don't really know what to do once I have the equation in the form $f(t,x,y) = 0$. How can this be calculated?
Edit: Image has been adjusted to better represent the given equation, all extra constants that were originally included have been set to unity.
Standard procedure C discriminant. Partially differentiate wrt $t$ and eliminate $t.$
Let a = -1000;
$$ e^{y/t} = \frac{at}{x}, \quad e^{y/t} \cdot \frac{-y}{t^2}=\frac{a}{x} \tag1 $$
Eliminate $e^{y/t} $ and cancell $\dfrac{a}{x}$
$$\to y=-t;\quad e^{-1}= \frac{-ay}{x} \tag 2 $$
$$\to \left( \frac{y}{x}= \frac{-1}{ae }\right)\tag 3 $$