If the line ` x + 2by + 7 = 0 ` is a diameter of the circle
` x^(2) + y^(2) - 6x + 2y = 0 `, then : b =

Options

A. 3
B. `- 5`
C. `-1`
D. 5

Answer

To find the value of \( b \) such that the line \( x + 2by + 7 = 0 \) is a diameter of the circle given by the equation \( x^2 + y^2 - 6x + 2y = 0 \), we can follow these steps:

### Step 1: Find the center of the circle

The equation of the circle is given as:
\[ x^2 + y^2 - 6x + 2y = 0 \]
We can rewrite this in standard form by completing the square.

1. Group the \( x \) and \( y \) terms:
\[ (x^2 - 6x) + (y^2 + 2y) = 0 \]

2. Complete the square for \( x \):
\[ x^2 - 6x = (x - 3)^2 - 9 \]

3. Complete the square for \( y \):
\[ y^2 + 2y = (y + 1)^2 - 1 \]

4. Substitute back into the equation:
\[ (x - 3)^2 - 9 + (y + 1)^2 - 1 = 0 \]
\[ (x - 3)^2 + (y + 1)^2 - 10 = 0 \]
\[ (x - 3)^2 + (y + 1)^2 = 10 \]

From this, we can see that the center of the circle is \( (3, -1) \).

### Step 2: Substitute the center into the line equation

The line \( x + 2by + 7 = 0 \) must pass through the center of the circle \( (3, -1) \). We substitute \( x = 3 \) and \( y = -1 \) into the line equation:

\[ 3 + 2b(-1) + 7 = 0 \]

### Step 3: Simplify the equation

Now, simplify the equation:
\[ 3 - 2b + 7 = 0 \]
\[ 10 - 2b = 0 \]

### Step 4: Solve for \( b \)

Now, solve for \( b \):
\[ 2b = 10 \]
\[ b = 5 \]

Thus, the value of \( b \) is \( 5 \).

### Final Answer
\[ b = 5 \]

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