The equation to sphere passing throrugh origin and the points (-1,0,0),(0,-2,0) and (0,0,-3) is `x^2+y^2+z^2+f(x,y,z)=0`. What if f(x,y,z) equal to ?

Options

A. `-x-2y-3z`
B. `x+2y+3z`
C. `x+2y+3z-1`
D. `x+2y+3z+1`

Explanation

To find the function \( f(x, y, z) \) in the equation of the sphere that passes through the origin and the points (-1, 0, 0), (0, -2, 0), and (0, 0, -3), we can follow these steps:

### Step 1: Understand the general equation of a sphere
The general equation of a sphere can be expressed as:
\[ x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0 \]
where \( (u, v, w) \) are constants that define the sphere.

### Step 2: Substitute the origin into the equation
Since the sphere passes through the origin (0, 0, 0), we substitute these coordinates into the equation:
\[ 0^2 + 0^2 + 0^2 + 2u(0) + 2v(0) + 2w(0) + d = 0 \]
This simplifies to:
\[ d = 0 \]

### Step 3: Substitute the first point (-1, 0, 0)
Next, we substitute the point (-1, 0, 0) into the equation:
\[ (-1)^2 + 0^2 + 0^2 + 2u(-1) + 2v(0) + 2w(0) + d = 0 \]
This simplifies to:
\[ 1 - 2u + 0 + 0 + 0 = 0 \implies 2u = 1 \implies u = \frac{1}{2} \]

### Step 4: Substitute the second point (0, -2, 0)
Now, we substitute the point (0, -2, 0):
\[ 0^2 + (-2)^2 + 0^2 + 2u(0) + 2v(-2) + 2w(0) + d = 0 \]
This simplifies to:
\[ 0 + 4 + 0 + 0 - 4v + 0 = 0 \implies 4 - 4v = 0 \implies v = 1 \]

### Step 5: Substitute the third point (0, 0, -3)
Finally, we substitute the point (0, 0, -3):
\[ 0^2 + 0^2 + (-3)^2 + 2u(0) + 2v(0) + 2w(-3) + d = 0 \]
This simplifies to:
\[ 0 + 0 + 9 + 0 + 0 - 6w + 0 = 0 \implies 9 - 6w = 0 \implies w = \frac{3}{2} \]

### Step 6: Write the function \( f(x, y, z) \)
Now that we have the values of \( u, v, w, \) and \( d \), we can express \( f(x, y, z) \):
\[ f(x, y, z) = 2ux + 2vy + 2wz + d \]
Substituting the values:
\[ f(x, y, z) = 2\left(\frac{1}{2}\right)x + 2(1)y + 2\left(\frac{3}{2}\right)z + 0 \]
This simplifies to:
\[ f(x, y, z) = x + 2y + 3z \]

### Final Answer
Thus, the function \( f(x, y, z) \) is:
\[ f(x, y, z) = x + 2y + 3z \]