If the lines `(x-1)/2=(y+1)/3=z/(5t-1)` and
`(x+1)/(2s+1)=y/2=z/4` are parallel to each other, then
value of s, t will be

Options

A. `6, 5/7`
B. `1/6, 7/5`
C. `3, 5/7`
D. `4, 7/10`

Answer

To solve the problem of finding the values of \( s \) and \( t \) such that the lines

\[ \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{5t-1} \]

and

\[ \frac{x+1}{2s+1} = \frac{y}{2} = \frac{z}{4} \]

are parallel, we can follow these steps:

### Step 1: Identify Direction Ratios
The direction ratios of the first line can be extracted from the equation:

\[ \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{5t-1} \]

This gives us the direction ratios \( \mathbf{d_1} = (2, 3, 5t - 1) \).

For the second line:

\[ \frac{x+1}{2s+1} = \frac{y}{2} = \frac{z}{4} \]

This gives us the direction ratios \( \mathbf{d_2} = (2s + 1, 2, 4) \).

### Step 2: Set Up Proportionality Condition
Since the lines are parallel, the direction ratios must be proportional. Therefore, we can set up the following equations based on the proportionality:

\[ \frac{2}{2s + 1} = \frac{3}{2} = \frac{5t - 1}{4} \]

### Step 3: Solve for \( s \)
From the first part of the proportionality:

\[ \frac{2}{2s + 1} = \frac{3}{2} \]

Cross-multiplying gives:

\[ 2 \cdot 2 = 3(2s + 1) \]

This simplifies to:

\[ 4 = 6s + 3 \]

Rearranging gives:

\[ 6s = 4 - 3 = 1 \implies s = \frac{1}{6} \]

### Step 4: Solve for \( t \)
Now, using the second part of the proportionality:

\[ \frac{3}{2} = \frac{5t - 1}{4} \]

Cross-multiplying gives:

\[ 3 \cdot 4 = 2(5t - 1) \]

This simplifies to:

\[ 12 = 10t - 2 \]

Rearranging gives:

\[ 10t = 12 + 2 = 14 \implies t = \frac{14}{10} = \frac{7}{5} \]

### Final Answer
Thus, the values of \( s \) and \( t \) are:

\[ s = \frac{1}{6}, \quad t = \frac{7}{5} \]