Consider the spheres `x^2+y^2+z^2-4y+3=0 and x^2+y^2+z^2+2x+4z-4=0`
What is the distance between the centres of the two spheres ?

Options

A. 5 unit
B. 4 unit
C. 3 unit
D. 2 unit

Answer

To find the distance between the centers of the two spheres given by the equations \(x^2 + y^2 + z^2 - 4y + 3 = 0\) and \(x^2 + y^2 + z^2 + 2x + 4z - 4 = 0\), we will follow these steps:

### Step 1: Rewrite the first sphere's equation
The first sphere's equation is:
\[ x^2 + y^2 + z^2 - 4y + 3 = 0 \]
We can rearrange this equation:
\[ x^2 + (y^2 - 4y) + z^2 + 3 = 0 \]
Next, we complete the square for the \(y\) term:
\[ y^2 - 4y = (y - 2)^2 - 4 \]
Substituting this back into the equation gives:
\[ x^2 + (y - 2)^2 - 4 + z^2 + 3 = 0 \]
This simplifies to:
\[ x^2 + (y - 2)^2 + z^2 - 1 = 0 \]
Thus, we have:
\[ x^2 + (y - 2)^2 + z^2 = 1 \]
From this, we can identify the center of the first sphere as:
\[ C_1(0, 2, 0) \]
and the radius \(r_1 = 1\).

### Step 2: Rewrite the second sphere's equation
The second sphere's equation is:
\[ x^2 + y^2 + z^2 + 2x + 4z - 4 = 0 \]
We rearrange this equation:
\[ (x^2 + 2x) + y^2 + (z^2 + 4z) - 4 = 0 \]
Next, we complete the square for the \(x\) and \(z\) terms:
\[ x^2 + 2x = (x + 1)^2 - 1 \]
\[ z^2 + 4z = (z + 2)^2 - 4 \]
Substituting these back into the equation gives:
\[ ((x + 1)^2 - 1) + y^2 + ((z + 2)^2 - 4) - 4 = 0 \]
This simplifies to:
\[ (x + 1)^2 + y^2 + (z + 2)^2 - 9 = 0 \]
Thus, we have:
\[ (x + 1)^2 + y^2 + (z + 2)^2 = 9 \]
From this, we can identify the center of the second sphere as:
\[ C_2(-1, 0, -2) \]
and the radius \(r_2 = 3\).

### Step 3: Calculate the distance between the centers
Now, we can calculate the distance \(d\) between the centers \(C_1(0, 2, 0)\) and \(C_2(-1, 0, -2)\) using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{((-1) - 0)^2 + (0 - 2)^2 + ((-2) - 0)^2} \]
This simplifies to:
\[ d = \sqrt{(-1)^2 + (-2)^2 + (-2)^2} \]
Calculating each term:
\[ d = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Final Answer
The distance between the centers of the two spheres is \(3\) units.

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