Find the equation of line which passes through a point where line given by `(x-1)/-1=(y_-2)/3=(z-4)/6` cuts the x axis and is parallel to the line `x/2=(y+1)/3=(z-2)/4`.

Answer

To find the equation of the line that passes through the point where the given line cuts the x-axis and is parallel to another line, we can follow these steps:

### Step 1: Identify the given line and its equation
The given line is represented by the equation:
\[ \frac{x-1}{-1} = \frac{y-2}{3} = \frac{z-4}{6} \]

### Step 2: Determine where the line cuts the x-axis
To find the point where this line intersects the x-axis, we set \(y = 0\) and \(z = 0\). Thus, we have:
\[ \frac{x-1}{-1} = \frac{0-2}{3} = \frac{0-4}{6} \]

### Step 3: Solve for \(x\)
From the equation \(\frac{0-2}{3}\), we get:
\[ \frac{x-1}{-1} = -\frac{2}{3} \]
This simplifies to:
\[ x - 1 = \frac{2}{3} \]
Adding 1 to both sides gives:
\[ x = \frac{2}{3} + 1 = \frac{2}{3} + \frac{3}{3} = \frac{5}{3} \]

### Step 4: Find the point of intersection
Thus, the point where the line cuts the x-axis is:
\[ \left(\frac{5}{3}, 0, 0\right) \]

### Step 5: Identify the direction ratios of the second line
The second line is given by:
\[ \frac{x}{2} = \frac{y+1}{3} = \frac{z-2}{4} \]
From this, we can extract the direction ratios as:
\[ (2, 3, 4) \]

### Step 6: Write the equation of the required line
Since the required line is parallel to the second line, it will have the same direction ratios. We can use the point \(\left(\frac{5}{3}, 0, 0\right)\) and the direction ratios \((2, 3, 4)\) to write the equation of the required line \(L_1\) in the symmetric form:
\[ \frac{x - \frac{5}{3}}{2} = \frac{y - 0}{3} = \frac{z - 0}{4} \]

### Final Equation
Thus, the equation of the required line is:
\[ \frac{x - \frac{5}{3}}{2} = \frac{y}{3} = \frac{z}{4} \]