Find the equation of a line passing though (2,0,5) and which is parallel line 6x-2=3y+1=2z-2

Answer

To find the equation of a line passing through the point (2, 0, 5) and parallel to the line given by the equation \(6x - 2 = 3y + 1 = 2z - 2\), we can follow these steps:

### Step 1: Convert the given line equation into parametric form
The given equation can be expressed in parametric form. We start by rewriting the equation:

\[ 6x - 2 = 3y + 1 = 2z - 2 \]

Let \(k\) be the parameter. We can express \(x\), \(y\), and \(z\) in terms of \(k\):

1. From \(6x - 2 = k\):
\[ 6x = k + 2 \implies x = \frac{k + 2}{6} \]

2. From \(3y + 1 = k\):
\[ 3y = k - 1 \implies y = \frac{k - 1}{3} \]

3. From \(2z - 2 = k\):
\[ 2z = k + 2 \implies z = \frac{k + 2}{2} \]

Thus, the parametric equations of the line are:
\[ x = \frac{k + 2}{6}, \quad y = \frac{k - 1}{3}, \quad z = \frac{k + 2}{2} \]

### Step 2: Identify the direction ratios of the given line
From the parametric equations, we can find the direction ratios of the line. The coefficients of \(k\) give us the direction ratios:

- For \(x\): Coefficient is \(\frac{1}{6}\)
- For \(y\): Coefficient is \(\frac{1}{3}\)
- For \(z\): Coefficient is \(\frac{1}{2}\)

Thus, the direction ratios of the line are \(1:2:3\).

### Step 3: Write the equation of the required line
The required line passes through the point \((2, 0, 5)\) and has the same direction ratios as the given line. Therefore, we can use the point-direction form of the line equation:

\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]

Substituting \(x_1 = 2\), \(y_1 = 0\), \(z_1 = 5\) and the direction ratios \(1, 2, 3\):

\[ \frac{x - 2}{1} = \frac{y - 0}{2} = \frac{z - 5}{3} \]

### Final Equation
Thus, the equation of the line is:

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