What is the meaning of $\delta(ax + by - t)$

Question

How one deals with 2D delta function $\delta(x+y)$?

Is $\delta (x+y)$ same as $\delta(x,y)=\delta(x) \delta(y)$ ?

It appears in radon transforms. Below is special case that I am interested in. $$\int\int \delta (ax+by-t) f(x)g(y) dx dy $$

Answer 1

No, it is simply a 1-D Dirac impulse being zero for non-zero argument, and satisfying

$$\int_{-\infty}^{\infty}f(x)\delta(x+y)dx=f(-y)$$