If`x^(2)+y^(2)=4 and a^(2)+b^(2)=9` then

Options

A. maximum value of (abxy)is 9
B. maximum value of (abxy)is 36
C. maximum value of (ax+by) is 6
D. minimum value of (ax+by) is -6

Answer

To solve the problem given, we need to analyze the equations provided and find the number of integer solutions for the equations \(x^2 + y^2 = 4\) and \(a^2 + b^2 = 9\).

### Step 1: Solve the first equation \(x^2 + y^2 = 4\)

We can find integer pairs \((x, y)\) that satisfy this equation. The possible integer values for \(x\) and \(y\) can be derived from the equation:

1. \(x = 0\):
- \(y^2 = 4 \Rightarrow y = \pm 2\)
- Solutions: \((0, 2)\), \((0, -2)\)

2. \(x = 1\):
- \(1^2 + y^2 = 4 \Rightarrow y^2 = 3\) (not an integer)

3. \(x = 2\):
- \(2^2 + y^2 = 4 \Rightarrow y^2 = 0 \Rightarrow y = 0\)
- Solution: \((2, 0)\)

4. \(x = -1\):
- \((-1)^2 + y^2 = 4 \Rightarrow y^2 = 3\) (not an integer)

5. \(x = -2\):
- \((-2)^2 + y^2 = 4 \Rightarrow y^2 = 0 \Rightarrow y = 0\)
- Solution: \((-2, 0)\)

Thus, the integer solutions for \(x^2 + y^2 = 4\) are:
- \((0, 2)\)
- \((0, -2)\)
- \((2, 0)\)
- \((-2, 0)\)

Total solutions: **4 pairs**.

### Step 2: Solve the second equation \(a^2 + b^2 = 9\)

Similarly, we find integer pairs \((a, b)\) that satisfy this equation:

1. \(a = 0\):
- \(b^2 = 9 \Rightarrow b = \pm 3\)
- Solutions: \((0, 3)\), \((0, -3)\)

2. \(a = 1\):
- \(1^2 + b^2 = 9 \Rightarrow b^2 = 8\) (not an integer)

3. \(a = 2\):
- \(2^2 + b^2 = 9 \Rightarrow b^2 = 5\) (not an integer)

4. \(a = 3\):
- \(3^2 + b^2 = 9 \Rightarrow b^2 = 0 \Rightarrow b = 0\)
- Solution: \((3, 0)\)

5. \(a = -1\):
- \((-1)^2 + b^2 = 9 \Rightarrow b^2 = 8\) (not an integer)

6. \(a = -2\):
- \((-2)^2 + b^2 = 9 \Rightarrow b^2 = 5\) (not an integer)

7. \(a = -3\):
- \((-3)^2 + b^2 = 9 \Rightarrow b^2 = 0 \Rightarrow b = 0\)
- Solution: \((-3, 0)\)

Thus, the integer solutions for \(a^2 + b^2 = 9\) are:
- \((0, 3)\)
- \((0, -3)\)
- \((3, 0)\)
- \((-3, 0)\)

Total solutions: **4 pairs**.

### Step 3: Combine the solutions

Now, we can combine the solutions from both equations. Each pair from the first equation can be combined with each pair from the second equation.

Total combinations = (Number of solutions for \(x^2 + y^2 = 4\)) × (Number of solutions for \(a^2 + b^2 = 9\)) = \(4 \times 4 = 16\).

### Final Answer

Thus, the total number of integer solutions for the given equations is **16**.