Calculate R1 and R2 given equivalent resistance

[SuperCat]

Note: I have come up with a solution myself, but I am trying to understand a different approach to the problem. My textbook solves it in a different manner, and I am having trouble following along.

Homework Statement

The last part of the problem "determine R1 and R2 such that Rin = 200k". I am having trouble understanding the solution for it. Note the graph for the problem is on the left.

Homework Equations

Ohm's law and KVL.

The Attempt at a Solution

Solution provided to me:

The solution for the resistance starts towards the bottom, once the gate voltage has been calculated. I do not understand why that formula is used to solve for the first resistor. To me it looks like conductance of the first resistor is being used with the Vdd to get a current. That current is being multiplied by the equivalent thevinin resistance of the two resistors. I just don't understand why that would be useful/how that would lead to calculating the resistance.

 

[SuperCat]

I have just come up with a solution myself. I took a very different approach but received the same answer.
Here is my approach:

I receive the correct answer, and I found this to be more straightforward. But I would appreciate if someone could explain to me how the textbook solution goes along.

 

[gneill]

Write two equations that involve R1 and R2:

1. The expression for the gate potential (voltage divider)
2. The expression for Rin

Look for commonalities in the two expressions that you can exploit.

 

[SuperCat]

Write two equations that involve R1 and R2:

1. The expression for the gate potential (voltage divider)
2. The expression for Rin

Look for commonalities in the two expressions that you can exploit.

Isn't that similar to what I have in the post above? I think my textbook does it with a current divider. Which gives me the impression that they had decided to assume there an AC signal, and decide to do small signal analysis.

 

[gneill]

Isn't that similar to what I have in the post above?

Yup. But then it's bound to be similar since there's not much leeway as to what are the "givens".

What I picked out and exploited was the fact that the term $\frac{R_2}{R_1 + R_2}$ occurs in both equations, and its occurrence in the voltage divider equation can be replaced with its value from the input resistance equation.