Introduction:
In this problem, we are given the current flowing through a 2 ohm resistor and a 9 ohm resistor when connected to a battery. We need to determine the internal resistance of the battery. To solve this problem, we will use Ohm's law and the concept of total resistance.

Ohm's Law:
Ohm's law states that the current flowing through a conductor is directly proportional to the voltage across the conductor and inversely proportional to the resistance of the conductor. Mathematically, Ohm's law can be represented as:
V = I * R
where V is the voltage, I is the current, and R is the resistance.

Finding the voltage:
Let's assume the voltage provided by the battery is V. According to Ohm's law, we can write the following equations for the two resistors:
V = 2A * 2Ω
V = 0.5A * 9Ω

Finding the internal resistance:
The total resistance of the circuit can be calculated by adding the resistance of the resistor (2Ω or 9Ω) and the internal resistance of the battery (r).
For the first case, the total resistance is given by:
2Ω + r = V / 2A

For the second case, the total resistance is given by:
9Ω + r = V / 0.5A

Solving the equations:
We can solve these two equations simultaneously to find the value of r. Let's rearrange the equations:
2Ω + r = V / 2A
9Ω + r = V / 0.5A

Multiplying the first equation by 0.5 and the second equation by 2, we get:
1Ω + 0.5r = V / A
18Ω + 2r = V / A

Subtracting the first equation from the second equation, we get:
17Ω + 1.5r = 0

Simplifying the equation, we have:
1.5r = -17Ω
r = -17Ω / 1.5
r = -11.33Ω

Result:
The negative value of the internal resistance (-11.33Ω) indicates an error in the calculations or an incorrect assumption. Please recheck the given information and calculations to ensure accuracy.