Find the centre and radius of sphere `2(x-5)(x+1)+2(y+5)(y-1)+2(z-2)(z+2)=7`.

Answer

To find the center and radius of the sphere given by the equation \(2(x-5)(x+1)+2(y+5)(y-1)+2(z-2)(z+2)=7\), we can follow these steps:

### Step 1: Simplify the Equation
Start with the given equation:
\[ 2(x-5)(x+1) + 2(y+5)(y-1) + 2(z-2)(z+2) = 7 \]
Factor out the 2:
\[ 2\left[(x-5)(x+1) + (y+5)(y-1) + (z-2)(z+2)\right] = 7 \]
Divide both sides by 2:
\[ (x-5)(x+1) + (y+5)(y-1) + (z-2)(z+2) = \frac{7}{2} \]

### Step 2: Expand Each Term
Now, expand each term:
1. For \( (x-5)(x+1) \):
\[ x^2 + x - 5x - 5 = x^2 - 4x - 5 \]
2. For \( (y+5)(y-1) \):
\[ y^2 - y + 5y - 5 = y^2 + 4y - 5 \]
3. For \( (z-2)(z+2) \):
\[ z^2 - 4 \]

Putting it all together:
\[ (x^2 - 4x - 5) + (y^2 + 4y - 5) + (z^2 - 4) = \frac{7}{2} \]

### Step 3: Combine Like Terms
Combine the terms:
\[ x^2 - 4x + y^2 + 4y + z^2 - 14 = \frac{7}{2} \]
Rearranging gives:
\[ x^2 - 4x + y^2 + 4y + z^2 = \frac{7}{2} + 14 \]
Convert \(14\) to a fraction:
\[ 14 = \frac{28}{2} \]
Thus,
\[ x^2 - 4x + y^2 + 4y + z^2 = \frac{35}{2} \]

### Step 4: Complete the Square
Now, we will complete the square for \(x\) and \(y\):
1. For \(x^2 - 4x\):
\[ (x-2)^2 - 4 \]
2. For \(y^2 + 4y\):
\[ (y+2)^2 - 4 \]

Substituting back into the equation:
\[ ((x-2)^2 - 4) + ((y+2)^2 - 4) + z^2 = \frac{35}{2} \]
This simplifies to:
\[ (x-2)^2 + (y+2)^2 + z^2 - 8 = \frac{35}{2} \]
Adding 8 to both sides:
\[ (x-2)^2 + (y+2)^2 + z^2 = \frac{35}{2} + 8 \]
Convert \(8\) to a fraction:
\[ 8 = \frac{16}{2} \]
Thus,
\[ (x-2)^2 + (y+2)^2 + z^2 = \frac{51}{2} \]

### Step 5: Identify the Center and Radius
The equation of the sphere is in the form:
\[ (x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 = r^2 \]
From this, we can identify:
- Center \((x_1, y_1, z_1) = (2, -2, 0)\)
- Radius \(r = \sqrt{\frac{51}{2}}\)

### Final Answer
The center of the sphere is \((2, -2, 0)\) and the radius is \(\sqrt{\frac{51}{2}}\).

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