Find the equation of the line passing through `((1)/(2) , -1,-(1)/(2))` and parallel to the line ` 6x - 2=3y +1= 2z-4`.

Answer

To find the equation of the line passing through the point \(\left(\frac{1}{2}, -1, -\frac{1}{2}\right)\) and parallel to the line given by the equations \(6x - 2 = 3y + 1 = 2z - 4\), we can follow these steps:

### Step 1: Identify the direction ratios of the given line
The given line is represented by the symmetric equations:
\[ 6x - 2 = 3y + 1 = 2z - 4 \]
We can rewrite this in a more standard form to find the direction ratios.

From \(6x - 2 = 3y + 1\), we have:
\[ 6x - 3y - 3 = 0 \quad \Rightarrow \quad 6x - 3y = 3 \]

From \(3y + 1 = 2z - 4\), we have:
\[ 3y - 2z + 5 = 0 \quad \Rightarrow \quad 3y - 2z = -5 \]

From \(6x - 2 = 2z - 4\), we have:
\[ 6x - 2z + 2 = 0 \quad \Rightarrow \quad 6x - 2z = -2 \]

The direction ratios can be derived from the coefficients of \(x\), \(y\), and \(z\). The direction ratios for the line are:
\[ (6, 3, 2) \]
To simplify, we can divide by 3:
\[ (2, 1, \frac{2}{3}) \]

### Step 2: Write the equation of the required line
Since the required line is parallel to the given line, it will have the same direction ratios. The point through which the line passes is \(\left(\frac{1}{2}, -1, -\frac{1}{2}\right)\).

Using the point-direction form of the line, we can express the equation as:
\[ \frac{x - \frac{1}{2}}{2} = \frac{y + 1}{1} = \frac{z + \frac{1}{2}}{\frac{2}{3}} \]

### Step 3: Convert to Cartesian form
To convert the symmetric form to Cartesian form, we can express each part as:
1. From \(\frac{x - \frac{1}{2}}{2} = t\), we have:
\[ x = 2t + \frac{1}{2} \]
2. From \(\frac{y + 1}{1} = t\), we have:
\[ y = t - 1 \]
3. From \(\frac{z + \frac{1}{2}}{\frac{2}{3}} = t\), we have:
\[ z = \frac{2}{3}t - \frac{1}{2} \]

### Final Equation
Thus, the equations of the line in parametric form are:
\[ x = 2t + \frac{1}{2}, \quad y = t - 1, \quad z = \frac{2}{3}t - \frac{1}{2} \]