The planes `ax+4y+z=0,2y+3z-1=0` and `3x-bz+2=0` will

Options

A. meet at a point if `ab ne 15`.
B. meet on a line if ab =15, a=3
C. have no common point if ab=15, `a ne 3`.
D. have no common point if ab=15, `ane5`

Answer

To solve the problem of determining the relationship between the planes given by the equations \( ax + 4y + z = 0 \), \( 2y + 3z - 1 = 0 \), and \( 3x - bz + 2 = 0 \), we will follow these steps:

### Step 1: Write the equations in standard form
The equations of the planes are:
1. \( ax + 4y + z = 0 \)
2. \( 2y + 3z - 1 = 0 \) (which can be rewritten as \( 0x + 2y + 3z = 1 \))
3. \( 3x - bz + 2 = 0 \) (which can be rewritten as \( 3x + 0y - bz = -2 \))

### Step 2: Set up the determinant
To find the condition for the planes to intersect, we can set up the determinant of the coefficients of \( x, y, z \):
\[ \begin{vmatrix} a & 4 & 1 \\ 0 & 2 & 3 \\ 3 & 0 & -b \end{vmatrix} \]

### Step 3: Calculate the determinant
Calculating the determinant:
\[ D = a \begin{vmatrix} 2 & 3 \\ 0 & -b \end{vmatrix} - 4 \begin{vmatrix} 0 & 3 \\ 3 & -b \end{vmatrix} + 1 \begin{vmatrix} 0 & 2 \\ 3 & 0 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \( \begin{vmatrix} 2 & 3 \\ 0 & -b \end{vmatrix} = 2(-b) - 0 = -2b \)
2. \( \begin{vmatrix} 0 & 3 \\ 3 & -b \end{vmatrix} = 0(-b) - 3(3) = -9 \)
3. \( \begin{vmatrix} 0 & 2 \\ 3 & 0 \end{vmatrix} = 0 - 6 = -6 \)

Substituting these back into the determinant:
\[ D = a(-2b) - 4(-9) + (-6) = -2ab + 36 - 6 = -2ab + 30 \]

### Step 4: Set the determinant equal to zero
For the planes to intersect at a point, the determinant must be equal to zero:
\[ -2ab + 30 = 0 \]
This simplifies to:
\[ 2ab = 30 \quad \Rightarrow \quad ab = 15 \]

### Step 5: Analyze the conditions
- If \( ab = 15 \), the planes intersect in a line.
- If \( ab \neq 15 \), the planes intersect at a point.

### Conclusion
Thus, the condition for the planes is:
- They intersect in a line if \( ab = 15 \).
- They intersect at a point if \( ab \neq 15 \).

### Final Answer
The correct answer is that the planes will intersect in a line if \( ab = 15 \).

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