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

Options

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

Answer

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 in standard form
The equations of the planes can be rewritten as follows:
1. Plane 1: \( bx - ay - n = 0 \)
2. Plane 2: \( cy - bz - l = 0 \)
3. Plane 3: \( az - cx - m = 0 \)

### Step 2: Use the condition for intersection of planes
The intersection of two planes can be expressed in terms of a linear combination of their equations. For planes to intersect in a line, the third plane must be expressible as a linear combination of the first two planes. This can be expressed as:
\[ P_1 + \lambda P_2 = 0 \]
where \( P_1 \) and \( P_2 \) are the equations of the first two planes, and \( \lambda \) is a scalar.

### Step 3: Substitute the plane equations
Substituting the equations of the planes into the linear combination gives:
\[ (bx - ay - n) + \lambda (cy - bz - l) = 0 \]
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 4: Compare coefficients
For the third plane \( az - cx - m = 0 \) to satisfy this equation, we need to compare coefficients of \( x \), \( y \), and \( z \) from both equations.

1. Coefficient of \( x \):
\[ b = -c \lambda \quad (1) \]

2. Coefficient of \( y \):
\[ \lambda c - a = 0 \quad (2) \]

3. Coefficient of \( z \):
\[ -\lambda b = a \quad (3) \]

4. Constant terms:
\[ -(n + \lambda l) = -m \quad (4) \]

### Step 5: Solve the equations
From equation (2):
\[ \lambda c = a \implies \lambda = \frac{a}{c} \quad (5) \]

Substituting (5) into equation (1):
\[ b = -c \left(\frac{a}{c}\right) \implies b = -a \]

Substituting (5) into equation (3):
\[ -\left(\frac{a}{c}\right)b = a \implies -\frac{ab}{c} = a \implies b = -c \quad (6) \]

Now substituting (5) into equation (4):
\[ -(n + \frac{a}{c}l) = -m \implies n + \frac{a}{c}l = m \]

### Step 6: Final condition
Rearranging gives us the final condition:
\[ c n + l a + b m = 0 \]

### Conclusion
The planes \( bx - ay = n \), \( cy - bz = l \), and \( az - cx = m \) intersect in a line if the condition \( c n + l a + b m = 0 \) holds.

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