Here is an update on my next attempt at this problem
I considered two cases
Case 1) ##v_A>e_2##
When this happens then we can treat the diode as a short circuit. We can analyze the resulting linear circuit using techniques for linear circuits, such as Thevenin method.
Here are the calculations
The resulting relationship between ##i_A## and ##v_A## is ##v_A=0.5+500i_A##.
Case 2: ##v_A<e_2##
We can treat the diode as a open circuit and again use Thevenin method.
The result is ##v_A=10^3i_A##.
Note that in the thick of things i used Thevenin but after the fact it is clear that that was unnecessary (as anuttarasammyak had previously shown).
Thus we have two different linear behaviors for this nonlinear "ideal diode model" diode.
Part b) of the problem inserts this non-linear two-terminal device in a circuit, as below
and asks for the load line.
It is a simple matter to obtain the load line.
$$\frac{v_T-3.5}{10^3}+i_T=0$$
and the load line is ##\frac{3.5-v_T}{10^3}##.
It is not clear to me, assuming everything above is correct, how to decide which case we are in, ie case 1 or case 2 of the diode.
If we are in case 1 then
$$\frac{v_T-3.5}{10^3}+\frac{v_T-0.5}{500}=0$$
and ##v_T=1.5V##V.
and if we are in case 2
$$\frac{v_T-3.5}{10^3}+10^{-3}v_T=0$$
and ##v_T=1.75##V.
This seems to indicate that this is not a possible solution for this circuit.
It seems to me that actually the threshold for the diode is 1V. For ##v_A## below 1V, the diode becomes an open circuit and ##e_2=1## and the entire NLD behaves like a linear resistor.
Once ##v_A## becomes ##>1##, now the diode changes behavior and becomes a linear circuit element with v-i characteristic ##v_A=0.5+500i_A##.
In the case of part b), for that circuit specifically, we have that ##v_A## (ie, ##v_T##) goes to ##1.5##V to balance out the equations.
The only weird thing to me is that I had to initially frame the cases in terms of the relationship of ##v_A## to ##e_2## but ultimately it seems that the cases are ##v_A>1## and ##v_A<1##.