If a and b are positive integers satisfying `(a+3)^(2)+(b+1)^(2)=85` , what is the minimum value of (2a+b)?

Options

A. 6
B. 7
C. 11
D. 12

Answer

To solve the problem, we start with the equation given:

\[ (a + 3)^2 + (b + 1)^2 = 85 \]

### Step 1: Identify the possible values for \(a + 3\) and \(b + 1\)

We can rewrite the equation as:

\[ x^2 + y^2 = 85 \]

where \(x = a + 3\) and \(y = b + 1\). We need to find pairs of integers \((x, y)\) such that their squares sum to 85.

### Step 2: List the pairs of integers

The integer pairs \((x, y)\) must satisfy \(x^2 + y^2 = 85\). We can check integer values for \(x\) from 0 to 9 (since \(9^2 = 81\) is the largest square less than 85):

- \(x = 0\): \(y^2 = 85\) (not an integer)
- \(x = 1\): \(y^2 = 84\) (not an integer)
- \(x = 2\): \(y^2 = 81 \Rightarrow y = 9\) \(\Rightarrow (2, 9)\)
- \(x = 3\): \(y^2 = 76\) (not an integer)
- \(x = 4\): \(y^2 = 69\) (not an integer)
- \(x = 5\): \(y^2 = 60\) (not an integer)
- \(x = 6\): \(y^2 = 49 \Rightarrow y = 7\) \(\Rightarrow (6, 7)\)
- \(x = 7\): \(y^2 = 36 \Rightarrow y = 6\) \(\Rightarrow (7, 6)\)
- \(x = 8\): \(y^2 = 21\) (not an integer)
- \(x = 9\): \(y^2 = 4 \Rightarrow y = 2\) \(\Rightarrow (9, 2)\)

The valid pairs are:
1. \((2, 9)\)
2. \((6, 7)\)
3. \((7, 6)\)
4. \((9, 2)\)

### Step 3: Calculate values of \(a\) and \(b\)

Now we convert these pairs back to \(a\) and \(b\):

1. For \((2, 9)\):
- \(a + 3 = 2 \Rightarrow a = -1\) (not valid since \(a\) is positive)
- \(b + 1 = 9 \Rightarrow b = 8\)

2. For \((6, 7)\):
- \(a + 3 = 6 \Rightarrow a = 3\)
- \(b + 1 = 7 \Rightarrow b = 6\)

3. For \((7, 6)\):
- \(a + 3 = 7 \Rightarrow a = 4\)
- \(b + 1 = 6 \Rightarrow b = 5\)

4. For \((9, 2)\):
- \(a + 3 = 9 \Rightarrow a = 6\)
- \(b + 1 = 2 \Rightarrow b = 1\)

### Step 4: Calculate \(2a + b\)

Now we calculate \(2a + b\) for valid pairs:

1. For \(a = -1\) and \(b = 8\): \(2(-1) + 8 = 6\) (not valid)
2. For \(a = 3\) and \(b = 6\): \(2(3) + 6 = 12\)
3. For \(a = 4\) and \(b = 5\): \(2(4) + 5 = 13\)
4. For \(a = 6\) and \(b = 1\): \(2(6) + 1 = 13\)

### Step 5: Find the minimum value

The valid values of \(2a + b\) are \(12\), \(13\), and \(13\). The minimum value is:

\[ \text{Minimum value of } (2a + b) = 12 \]

### Final Answer

The minimum value of \(2a + b\) is \(12\).

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