If the centre of the sphere
`ax^2+by^2+cx^2-2x+4y+2z-3=0" is " (1//2,-1,1//2)`, what is the value of b?

Options

A. 1
B. -1
C. 2
D. -2

Explanation

To find the value of \( b \) in the equation of the sphere given by

\[ ax^2 + by^2 + cz^2 - 2x + 4y + 2z - 3 = 0, \]

with the center of the sphere specified as \( \left( \frac{1}{2}, -1, \frac{1}{2} \right) \), we can follow these steps:

### Step 1: Identify the general form of the sphere's equation
The general equation of a sphere can be expressed in the form:

\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0, \]

where \( (u, v, w) \) represents the coordinates of the center of the sphere.

### Step 2: Rewrite the given equation in a comparable form
Given the equation:

\[ ax^2 + by^2 + cz^2 - 2x + 4y + 2z - 3 = 0, \]

we know that for a sphere, the coefficients of \( x^2 \), \( y^2 \), and \( z^2 \) must be equal. Therefore, we can set \( a = b = c = k \) for some constant \( k \). Thus, we rewrite the equation as:

\[ kx^2 + ky^2 + kz^2 - 2x + 4y + 2z - 3 = 0. \]

### Step 3: Divide the entire equation by \( k \)
To simplify, we divide the entire equation by \( k \):

\[ x^2 + y^2 + z^2 - \frac{2}{k}x + \frac{4}{k}y + \frac{2}{k}z - \frac{3}{k} = 0. \]

### Step 4: Identify coefficients for center coordinates
From the general form of the sphere's equation, we can identify:

- \( 2u = -\frac{2}{k} \) → \( u = -\frac{1}{k} \)
- \( 2v = \frac{4}{k} \) → \( v = \frac{2}{k} \)
- \( 2w = \frac{2}{k} \) → \( w = \frac{1}{k} \)

### Step 5: Write down the center of the sphere
The center of the sphere is given by \( (-u, -v, -w) \), which gives us:

\[ \left( \frac{1}{k}, -\frac{2}{k}, -\frac{1}{k} \right). \]

### Step 6: Set the center equal to the given center
We know the center of the sphere is \( \left( \frac{1}{2}, -1, \frac{1}{2} \right) \). Thus, we can set up the following equations:

1. \( \frac{1}{k} = \frac{1}{2} \)
2. \( -\frac{2}{k} = -1 \)
3. \( -\frac{1}{k} = \frac{1}{2} \)

### Step 7: Solve for \( k \)
From the first equation:

\[ \frac{1}{k} = \frac{1}{2} \implies k = 2. \]

### Step 8: Determine the value of \( b \)
Since we established that \( a = b = c = k \), we have:

\[ b = k = 2. \]

### Final Answer
Thus, the value of \( b \) is

\[ \boxed{2}. \]