The distance of the centre of the sphere `x^2 + y^2 + z^2 - 2x - 4y = 0` from the origin is

Options

A. 5
B. `sqrt5`
C. `2sqrt5`
D. `5/2`

Answer

To find the distance of the center of the sphere given by the equation \(x^2 + y^2 + z^2 - 2x - 4y = 0\) from the origin, we can follow these steps:

### Step 1: Rewrite the equation of the sphere
The equation of the sphere can be rearranged in the standard form. The given equation is:
\[ x^2 + y^2 + z^2 - 2x - 4y = 0 \]
We can rewrite this as:
\[ x^2 - 2x + y^2 - 4y + z^2 = 0 \]

### Step 2: Complete the square for \(x\) and \(y\)
To find the center of the sphere, we need to complete the square for the \(x\) and \(y\) terms.

For \(x^2 - 2x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]

For \(y^2 - 4y\):
\[ y^2 - 4y = (y - 2)^2 - 4 \]

Substituting these back into the equation gives:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 + z^2 = 0 \]
This simplifies to:
\[ (x - 1)^2 + (y - 2)^2 + z^2 = 5 \]

### Step 3: Identify the center of the sphere
From the standard form of the sphere \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), we can identify the center \((h, k, l)\) and the radius \(r\). Here, the center is:
\[ (1, 2, 0) \]

### Step 4: Calculate the distance from the origin
The distance \(d\) from the center of the sphere \((1, 2, 0)\) to the origin \((0, 0, 0)\) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of the center and the origin:
\[ d = \sqrt{(1 - 0)^2 + (2 - 0)^2 + (0 - 0)^2} \]
This simplifies to:
\[ d = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Final Answer
The distance of the center of the sphere from the origin is \(\sqrt{5}\).

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