The equation of a line passing through (a, b, c)
and parallel tp z-axis is

Options

A. `(x-a)/1=(y-b)/1=(z-c)/0`
B. `(x-a)/0=(y-b)/1=(z-c)/1`
C. `(x-a)/1=(y-b)/1=(z-c)/1`
D. `(x-a)/0=(y-b)/0=(z-c)/1`

Answer

To find the equation of a line that passes through the point \((a, b, c)\) and is parallel to the z-axis, we can follow these steps:

### Step 1: Understand the general equation of a line in 3D
The equation of a line in three-dimensional space that passes through a point \((x_1, y_1, z_1)\) and has direction ratios \(l, m, n\) is given by:
\[ \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} \]

### Step 2: Identify the point and direction ratios
In this case, the line passes through the point \((a, b, c)\) and is parallel to the z-axis. The direction ratios for a line parallel to the z-axis are:
\[ l = 0, \quad m = 0, \quad n = 1 \]

### Step 3: Substitute the point and direction ratios into the equation
Substituting \(x_1 = a\), \(y_1 = b\), \(z_1 = c\), and the direction ratios \(l = 0\), \(m = 0\), \(n = 1\) into the general equation, we get:
\[ \frac{x - a}{0} = \frac{y - b}{0} = \frac{z - c}{1} \]

### Step 4: Simplify the equation
The terms \(\frac{x - a}{0}\) and \(\frac{y - b}{0}\) are undefined, which indicates that \(x\) and \(y\) can take any value. Therefore, we can express the equations as:
\[ x = a \quad \text{and} \quad y = b \]
And since the line is parallel to the z-axis, \(z\) can take any value. Thus, we can write:
\[ z = c + t \quad \text{where } t \text{ is a parameter} \]

### Final Equation
Combining these, the parametric equations of the line can be expressed as:
\[ x = a, \quad y = b, \quad z = c + t \]

### Conclusion
The equation of the line passing through the point \((a, b, c)\) and parallel to the z-axis is:
\[ x = a, \quad y = b, \quad z \text{ is free} \]