Solving circuit design problem with two approaches

Question

A voltage of 200V is applied to a tapped resistor of 500 ohm. Find the resistance between two tapping points connected to a circuit needing 0.1A at 25V. 
I have solved this problem with two approach. But the correct answer is coming only by one approach.

Total resistance is Ra+Rb = 500 ohm. 
Va+Vb = 200V ; Vb = 25V ;  
Hence Va = 175V.
Applying KCL at the tapping point we get:
-I + 0.1 + 25/Rb = 0 ; I = 0.1 + 25/Rb; 
Va = I.Ra; 
175 = (0.1 + 25/Rb)(500-Rb)
If we solve we will get Ra = 420.85 ohm and Rb = 79.15 ohm.

So far so good.

Consider second image in which Ra and Rb variable are taken such that Ra + Rb = 500 ohm 
Va = 175V 
Applying KCL at the tapping point we get:
-I + 0.1 + 25/(500-Ra) = 0 ; I = 0.1 + 25/500-Ra; 
Va = I.Ra; 
175 = (0.1 + 25/500-Ra)(Ra)
If we solve we will get Ra = 2079.15 ohm. 
which is wrong, but I am unable to understand the flaw in the second case. It is perfectly fine. The answer is not correct. Just by changing the way variable are taken the answer is changing.

po.pe · Answer

The flaw is that this equation has some restrictions when applied to a real circuit. From a mathematical perspective the second solution is correct. But as still applied that

$Ra=500Omega-Rb$

you would end up with a negative resistance for

$Ra = -1579Omega$

In the real world such a thing doesn't exist and therefore the solution is not applicable for your problem.

Solving circuit design problem with two approaches

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