If `x^(2) + y ^(2) - 2x + 6y + 10 =0` then the value of `(x ^(2) + y ^(2))` is

Options

A. 4
B. 6
C. 8
D. 10

Answer

To solve the equation \( x^2 + y^2 - 2x + 6y + 10 = 0 \) and find the value of \( x^2 + y^2 \), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the equation to isolate the terms involving \( x \) and \( y \):
\[ x^2 - 2x + y^2 + 6y + 10 = 0 \]

### Step 2: Complete the square for \( x \)
To complete the square for the \( x \) terms, take the coefficient of \( x \), which is \(-2\), halve it to get \(-1\), and then square it to get \(1\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]

### Step 3: Complete the square for \( y \)
Next, complete the square for the \( y \) terms. The coefficient of \( y \) is \(6\). Halve it to get \(3\) and square it to get \(9\):
\[ y^2 + 6y = (y + 3)^2 - 9 \]

### Step 4: Substitute back into the equation
Now substitute the completed squares back into the equation:
\[ ((x - 1)^2 - 1) + ((y + 3)^2 - 9) + 10 = 0 \]
This simplifies to:
\[ (x - 1)^2 + (y + 3)^2 - 1 - 9 + 10 = 0 \]
\[ (x - 1)^2 + (y + 3)^2 = 0 \]

### Step 5: Analyze the equation
The equation \( (x - 1)^2 + (y + 3)^2 = 0 \) implies that both squares must equal zero:
\[ x - 1 = 0 \quad \text{and} \quad y + 3 = 0 \]
Thus, we find:
\[ x = 1 \quad \text{and} \quad y = -3 \]

### Step 6: Calculate \( x^2 + y^2 \)
Now we can find \( x^2 + y^2 \):
\[ x^2 + y^2 = 1^2 + (-3)^2 = 1 + 9 = 10 \]

### Final Answer
The value of \( x^2 + y^2 \) is \( \boxed{10} \).

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