If `x=ay-1=z-2, and x=3y-2=bz-2` lie in same plane then the value of a and b is

Options

A. a=2,b=3
B. a=1,b=1
C. `b=1,a inR-{0}`
D. a=3,b=2

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) such that the lines given by the equations \( x = ay - 1 = z - 2 \) and \( x = 3y - 2 = bz - 2 \) lie in the same plane.

### Step-by-Step Solution:

1. **Rewrite the equations in standard form:**
- For the first line:
\[ x = ay - 1 = z - 2 \]
We can express this as:
\[ \frac{x}{1} = \frac{y - 1}{a} = \frac{z - 2}{1} \]
- For the second line:
\[ x = 3y - 2 = bz - 2 \]
This can be expressed as:
\[ \frac{x}{1} = \frac{y - \frac{2}{3}}{1/3} = \frac{z - 2}{1/b} \]

2. **Identify the parameters from the standard form:**
- From the first line, we have:
- \( x_1 = 0, y_1 = 1, z_1 = 2 \)
- Direction ratios: \( a_1 = 1, b_1 = a, c_1 = 1 \)
- From the second line, we have:
- \( x_2 = 0, y_2 = \frac{2}{3}, z_2 = 2 \)
- Direction ratios: \( a_2 = 1, b_2 = 3, c_2 = b \)

3. **Set up the determinant condition for coplanarity:**
The lines are coplanar if the determinant of the following matrix is zero:
\[ \begin{vmatrix} x_1 - x_2 & y_1 - y_2 & z_1 - z_2 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = 0 \]
Substituting the values:
\[ \begin{vmatrix} 0 - 0 & 1 - \frac{2}{3} & 2 - 2 \\ 1 & a & 1 \\ 1 & 3 & b \end{vmatrix} = 0 \]
This simplifies to:
\[ \begin{vmatrix} 0 & \frac{1}{3} & 0 \\ 1 & a & 1 \\ 1 & 3 & b \end{vmatrix} = 0 \]

4. **Calculate the determinant:**
The determinant can be computed as follows:
\[ 0 \cdot (a \cdot b - 3) - \frac{1}{3} \cdot (1 \cdot b - 1 \cdot 3) + 0 = 0 \]
This simplifies to:
\[ -\frac{1}{3} (b - 3) = 0 \]
Thus, we have:
\[ b - 3 = 0 \implies b = 3 \]

5. **Finding the value of \( a \):**
Since \( a \) does not appear in the determinant condition, it can take any real value except for zero (to avoid division by zero in the direction ratios). Thus, \( a \) can be any real number \( a \neq 0 \).

### Final Answer:
The values of \( a \) and \( b \) are:
\[ a \text{ is any real number } \neq 0, \quad b = 3 \]