The number of positive integral solutions `(x,y)` of the equation `2xy-4x^(2)+12x-5y=11,` is

Options

A. 0
B. 1
C. 2
D. infinitely many

Answer

To find the number of positive integral solutions \((x, y)\) of the equation \(2xy - 4x^2 + 12x - 5y = 11\), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ 2xy - 4x^2 + 12x - 5y = 11 \]
Rearranging gives:
\[ 2xy - 5y = 4x^2 - 12x + 11 \]

### Step 2: Factoring Out \(y\)
Factor out \(y\) from the left side:
\[ y(2x - 5) = 4x^2 - 12x + 11 \]
Now, we can express \(y\) in terms of \(x\):
\[ y = \frac{4x^2 - 12x + 11}{2x - 5} \]

### Step 3: Finding Conditions for \(y\) to be Positive
For \(y\) to be a positive integer, the right-hand side must be a positive integer. This requires:
1. \(2x - 5 \neq 0\) (to avoid division by zero).
2. \(4x^2 - 12x + 11\) must be divisible by \(2x - 5\).

### Step 4: Polynomial Long Division
Perform polynomial long division of \(4x^2 - 12x + 11\) by \(2x - 5\):
1. Divide \(4x^2\) by \(2x\) to get \(2x\).
2. Multiply \(2x\) by \(2x - 5\) to get \(4x^2 - 10x\).
3. Subtract:
\[ (4x^2 - 12x + 11) - (4x^2 - 10x) = -2x + 11 \]
4. Divide \(-2x\) by \(2x\) to get \(-1\).
5. Multiply \(-1\) by \(2x - 5\) to get \(-2x + 5\).
6. Subtract:
\[ (-2x + 11) - (-2x + 5) = 6 \]
Thus, we have:
\[ \frac{4x^2 - 12x + 11}{2x - 5} = 2x - 1 + \frac{6}{2x - 5} \]

### Step 5: Ensuring \(y\) is Positive
For \(y\) to be a positive integer:
\[ \frac{6}{2x - 5} \text{ must be a positive integer.} \]
This means \(2x - 5\) must be a divisor of 6.

### Step 6: Finding Divisors of 6
The positive divisors of 6 are \(1, 2, 3, 6\). We can set up equations:
1. \(2x - 5 = 1 \Rightarrow 2x = 6 \Rightarrow x = 3\)
2. \(2x - 5 = 2 \Rightarrow 2x = 7 \Rightarrow x = 3.5\) (not an integer)
3. \(2x - 5 = 3 \Rightarrow 2x = 8 \Rightarrow x = 4\)
4. \(2x - 5 = 6 \Rightarrow 2x = 11 \Rightarrow x = 5.5\) (not an integer)

### Step 7: Finding Corresponding \(y\) Values
Now, we substitute the valid \(x\) values back into the equation to find \(y\):
1. For \(x = 3\):
\[ y = 2(3) - 1 + \frac{6}{1} = 6 - 1 + 6 = 11 \]
2. For \(x = 4\):
\[ y = 2(4) - 1 + \frac{6}{3} = 8 - 1 + 2 = 9 \]

### Step 8: Conclusion
The positive integral solutions \((x, y)\) are \((3, 11)\) and \((4, 9)\). Thus, the number of positive integral solutions is:
\[ \boxed{2} \]