Under what condition do the planes `bx-ay=n,cy-bz=l,az-cx=m` intersect in a line?

Options

A. `a+b+c=0`
B. `a=b=c`
C. `al+bm+cn=0`
D. `l+m+n=0`

Explanation

To determine the condition under which the planes \( bx - ay = n \), \( cy - bz = l \), and \( az - cx = m \) intersect in a line, we can follow these steps:

### Step 1: Write the equations of the planes
We denote the planes as follows:
- Plane 1: \( P_1: bx - ay - n = 0 \)
- Plane 2: \( P_2: cy - bz - l = 0 \)
- Plane 3: \( P_3: az - cx - m = 0 \)

### Step 2: Set up the equation for the intersection of two planes
The intersection of two planes can be expressed as:
\[ P_1 + \lambda P_2 = 0 \]
This gives us:
\[ (bx - ay - n) + \lambda (cy - bz - l) = 0 \]
Expanding this, we have:
\[ bx - ay - n + \lambda cy - \lambda bz - \lambda l = 0 \]
Rearranging, we get:
\[ bx + (\lambda c - a)y - \lambda bz - (n + \lambda l) = 0 \]

### Step 3: For the planes to intersect in a line, the third plane must be satisfied
We need the equation of the third plane \( P_3 \) to hold true:
\[ az - cx - m = 0 \]

### Step 4: Compare coefficients
We can compare coefficients of \( x \), \( y \), \( z \), and the constant terms from the combined equation \( P_1 + \lambda P_2 \) with \( P_3 \).

1. **Coefficient of \( y \)**:
\[ -a + \lambda c = 0 \implies \lambda c = a \implies \lambda = \frac{a}{c} \]

2. **Coefficient of \( x \)**:
\[ b = -\lambda c \implies b = -\frac{a}{c}c \implies b = -a \]

3. **Coefficient of \( z \)**:
\[ a = -\lambda b \implies a = -\frac{a}{c}b \]
Substituting \( b = -a \):
\[ a = -\frac{a}{c}(-a) \implies a = \frac{a^2}{c} \implies c = \frac{a^2}{a} = a \]

4. **Constant terms**:
\[ -n = -\lambda l \implies n = \lambda l \implies n = \frac{a}{c}l \]

### Step 5: Combine results
From the above comparisons, we have:
- \( b = -a \)
- \( c = a \)
- \( n = \frac{a}{c}l \)

### Final Condition
Putting these together, we find that the planes intersect in a line if:
\[ AL + MB + CN = 0 \]
where \( A = a, B = b, C = c \).