The equation of the sphere concentric with the sphere `x^(2)+y^(2)+z^(2)-2x-6y-8z-5=0` and which passes through the origin is

Options

A. `x^(2)+y^(2)+z^(2)-2x-6y-8z=0`
B. `x^(2)+y^(2)+z^(2)-6y-8z=0`
C. `x^(2)+y^(2)+z^(2)=0`
D. None of these

Answer

To find the equation of the sphere that is concentric with the given sphere and passes through the origin, we can follow these steps:

### Step 1: Identify the given sphere's equation
The equation of the given sphere is:
\[ x^2 + y^2 + z^2 - 2x - 6y - 8z - 5 = 0 \]

### Step 2: Rewrite the equation in standard form
We can rewrite the equation of the sphere in standard form by completing the square for the \(x\), \(y\), and \(z\) terms.

1. For \(x\):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
2. For \(y\):
\[ y^2 - 6y = (y - 3)^2 - 9 \]
3. For \(z\):
\[ z^2 - 8z = (z - 4)^2 - 16 \]

Substituting these back into the equation gives:
\[ (x - 1)^2 - 1 + (y - 3)^2 - 9 + (z - 4)^2 - 16 - 5 = 0 \]
Simplifying this:
\[ (x - 1)^2 + (y - 3)^2 + (z - 4)^2 - 31 = 0 \]
Thus, we have:
\[ (x - 1)^2 + (y - 3)^2 + (z - 4)^2 = 31 \]
This shows that the center of the sphere is at \((1, 3, 4)\) and the radius is \(\sqrt{31}\).

### Step 3: Write the general equation of a concentric sphere
The general equation of a sphere concentric with the given sphere can be written as:
\[ x^2 + y^2 + z^2 + 2gx + 2fy + 2hz + d = 0 \]
where \((g, f, h)\) are the coordinates of the center of the sphere. Since the new sphere is concentric with the original sphere, it will have the same center \((1, 3, 4)\).

Thus, we have:
\[ g = -1, \quad f = -3, \quad h = -4 \]

### Step 4: Determine the value of \(d\) for the sphere passing through the origin
Since the sphere passes through the origin \((0, 0, 0)\), we substitute these coordinates into the general equation to find \(d\):
\[ 0^2 + 0^2 + 0^2 + 2(-1)(0) + 2(-3)(0) + 2(-4)(0) + d = 0 \]
This simplifies to:
\[ d = 0 \]

### Step 5: Write the equation of the required sphere
Substituting \(g\), \(f\), \(h\), and \(d\) back into the sphere's equation gives:
\[ x^2 + y^2 + z^2 - 2x - 6y - 8z = 0 \]

### Final Answer
The equation of the sphere concentric with the given sphere and passing through the origin is:
\[ x^2 + y^2 + z^2 - 2x - 6y - 8z = 0 \]