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Suppose we have a double delta function:

$$\delta(ax+by+c) \delta(Ax + By + C)$$

where $ax+by+c=0$ and $Ax + By + C=0$ has a solution $x=x_0, y=y_0$. Is it possible to simplify the double delta function to something like

$$\delta(ax+by+c) \delta(Ax + By + C) = \text{(some factor)}\times \delta(x-x_0)\delta(y-y_0)$$

If so, what is the factor?

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This is a well-defined distribution, and its action on $\phi$ can be evaluated as an iterated integral. Assuming $b = B = -1$, $$(f, \phi) = \iint_{\mathbb R^2} \delta(y - k_1 (x - x_0) - y_0) \,\delta(y - k_2 (x - x_0) - y_0) \,\phi(x, y) \,dy dx = \\ \int_{\mathbb R} \delta((k_2 - k_1) (x - x_0)) \,\phi(x, k_2 (x - x_0) + y_0) \,dx = \frac 1 {|k_1 - k_2|} \phi(x_0, y_0), \\ f = \frac 1 {|k_1 - k_2|} \delta(x - x_0) \delta(y - y_0).$$

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