The equation of the sphere circumscribing the tetrahedron whose faces are `x=0,y=0,z=0` and `x/a+y/b+z/c=1`, is equal to

Options

A. `x^(2)+y^(2)+z^(2)=a^(2)+b^(2)+c^(2)`
B. `x^(2)+y^(2)+z^(2)-ax+by+cz=0`
C. `x^(2)+y^(2)+z^(2)-2ax+2by+2cz=0`
D. none of these

Answer

To find the equation of the sphere circumscribing the tetrahedron defined by the planes \(x=0\), \(y=0\), \(z=0\), and \(x/a + y/b + z/c = 1\), we will follow these steps:

### Step 1: Identify the vertices of the tetrahedron
The tetrahedron is formed by the intersection of the given planes. The vertices can be found as follows:
- The intersection of the planes \(x=0\), \(y=0\), and \(z=0\) gives the vertex \(O(0, 0, 0)\).
- The intersection of the plane \(x/a + y/b + z/c = 1\) with the coordinate planes gives the other vertices:
- Setting \(y=0\) and \(z=0\) gives \(A(a, 0, 0)\).
- Setting \(x=0\) and \(z=0\) gives \(B(0, b, 0)\).
- Setting \(x=0\) and \(y=0\) gives \(C(0, 0, c)\).

Thus, the vertices of the tetrahedron are \(O(0, 0, 0)\), \(A(a, 0, 0)\), \(B(0, b, 0)\), and \(C(0, 0, c)\).

### Step 2: Write the general equation of the sphere
The general equation of a sphere in three-dimensional space is given by:
\[ x^2 + y^2 + z^2 + ux + vy + wz + d = 0 \]
Since the sphere passes through the origin \(O(0, 0, 0)\), we can simplify the equation to:
\[ x^2 + y^2 + z^2 + ux + vy + wz = 0 \]

### Step 3: Use the vertices to find coefficients
We will substitute the coordinates of the vertices into the sphere's equation to find the coefficients \(u\), \(v\), and \(w\).

1. **Substituting vertex \(A(a, 0, 0)\)**:
\[ a^2 + ua = 0 \implies u = -\frac{a^2}{a} = -a \]

2. **Substituting vertex \(B(0, b, 0)\)**:
\[ b^2 + vb = 0 \implies v = -\frac{b^2}{b} = -b \]

3. **Substituting vertex \(C(0, 0, c)\)**:
\[ c^2 + wc = 0 \implies w = -\frac{c^2}{c} = -c \]

### Step 4: Substitute \(u\), \(v\), and \(w\) back into the sphere's equation
Now we substitute \(u\), \(v\), and \(w\) back into the sphere's equation:
\[ x^2 + y^2 + z^2 - ax - by - cz = 0 \]

### Final Equation of the Sphere
Thus, the equation of the sphere circumscribing the tetrahedron is:
\[ x^2 + y^2 + z^2 - ax - by - cz = 0 \]