The shortest distance between any two opposite edges of the tetrahedron formed by planes `x+y=0, y+z=0, z+x=0, x+y+z=a` is constant, equal to

Options

A. `2a`
B. `(2a)/(sqrt(6))`
C. `(a)/(sqrt(6))`
D. `(2a)/(sqrt(3))`

Answer

To find the shortest distance between any two opposite edges of the tetrahedron formed by the planes \(x+y=0\), \(y+z=0\), \(z+x=0\), and \(x+y+z=a\), we can follow these steps:

### Step 1: Identify the vertices of the tetrahedron
The planes intersect to form a tetrahedron. To find the vertices, we can solve the equations of the planes in pairs.

1. **Intersection of \(x+y=0\) and \(y+z=0\)**:
- Let \(z=t\). Then, \(y=-t\) and \(x=t\).
- This gives us the point \(A(t, -t, t)\).

2. **Intersection of \(y+z=0\) and \(z+x=0\)**:
- Let \(x=p\). Then, \(z=-p\) and \(y=p\).
- This gives us the point \(B(p, p, -p)\).

3. **Intersection of \(z+x=0\) and \(x+y+z=a\)**:
- Let \(y=q\). Then, \(x=-q\) and \(z=a+q\).
- This gives us the point \(C(-q, q, a+q)\).

4. **Intersection of \(x+y=0\) and \(x+y+z=a\)**:
- Let \(z=r\). Then, \(x=-\frac{a-r}{2}\) and \(y=\frac{a-r}{2}\).
- This gives us the point \(D\left(-\frac{a-r}{2}, \frac{a-r}{2}, r\right)\).

### Step 2: Find the parametric equations for the edges
Using the vertices found, we can express the edges of the tetrahedron in parametric form. For example, the edge between points \(A\) and \(B\) can be expressed as:

- \(AB: (t, -t, t) + k\left(p-t, p+t, -t\right)\) where \(k\) is a parameter.

### Step 3: Calculate the distance between opposite edges
To find the shortest distance between opposite edges, we can use the formula for the distance between two skew lines represented in parametric form. The distance \(d\) between two lines can be calculated as:

\[ d = \frac{|(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|} \]

Where:
- \(\vec{a_1}\) and \(\vec{a_2}\) are points on the two lines.
- \(\vec{b_1}\) and \(\vec{b_2}\) are direction vectors of the two lines.

### Step 4: Substitute and simplify
1. Identify the direction vectors and points for the edges.
2. Substitute into the distance formula.
3. Simplify the expression to find the shortest distance.

### Step 5: Find the minimum distance
To find the minimum distance, differentiate the distance function with respect to the parameter and set the derivative to zero. Solve for the parameter to find the critical points, and evaluate the distance at these points.

### Final Result
After performing the calculations, we find that the shortest distance \(d\) between the opposite edges of the tetrahedron is given by:

\[ d = \frac{2a}{\sqrt{6}} \]