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 determine 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 need to compare their direction ratios.

### Step 1: Identify the direction ratios of the first line

From the first line, we can express it in the form of direction ratios. The direction ratios can be derived from the coefficients of \( x \), \( y \), and \( z \):

- For the first line, the direction ratios are \( (2, 3, 5t - 1) \).

### Step 2: Identify the direction ratios of the second line

Similarly, for the second line, we can express it in the same manner:

- For the second line, the direction ratios are \( (2s + 1, 2, 4) \).

### Step 3: Set up the proportionality condition

Since the lines are parallel, their direction ratios must be proportional. This means:

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

### Step 4: Solve for \( s \)

From the first part of the proportion:

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

Cross-multiplying gives:

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

This simplifies to:

\[ 4 = 6s + 3 \]

Rearranging gives:

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

### Step 5: Solve for \( t \)

Now, using the second part of the proportion:

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

Cross-multiplying gives:

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

This simplifies to:

\[ 12 = 10t - 2 \]

Rearranging gives:

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

### Final Answer

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

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