The arithmetic mean of two numbers x and y is `3` and geometric mean is `1`. Then `x^(2)+y^(2)` is equal to

Options

A. 30
B. 31
C. 32
D. 34

Answer

To solve the problem, we need to find \( x^2 + y^2 \) given that the arithmetic mean of \( x \) and \( y \) is 3 and the geometric mean is 1.

### Step-by-Step Solution:

1. **Understanding Arithmetic Mean**:
The arithmetic mean of two numbers \( x \) and \( y \) is given by:
\[ \text{AM} = \frac{x + y}{2} \]
Given that the arithmetic mean is 3, we can set up the equation:
\[ \frac{x + y}{2} = 3 \]

2. **Solving for \( x + y \)**:
To find \( x + y \), we multiply both sides of the equation by 2:
\[ x + y = 6 \quad \text{(Equation 1)} \]

3. **Understanding Geometric Mean**:
The geometric mean of two numbers \( x \) and \( y \) is given by:
\[ \text{GM} = \sqrt{xy} \]
Given that the geometric mean is 1, we can set up the equation:
\[ \sqrt{xy} = 1 \]

4. **Solving for \( xy \)**:
Squaring both sides of the equation gives us:
\[ xy = 1 \quad \text{(Equation 2)} \]

5. **Using the Formula for \( x^2 + y^2 \)**:
We can use the identity:
\[ x^2 + y^2 = (x + y)^2 - 2xy \]
Substituting the values from Equation 1 and Equation 2 into this formula:
\[ x^2 + y^2 = (6)^2 - 2(1) \]

6. **Calculating \( x^2 + y^2 \)**:
Now we calculate:
\[ x^2 + y^2 = 36 - 2 = 34 \]

### Final Answer:
Thus, the value of \( x^2 + y^2 \) is \( \boxed{34} \).

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