What is the equation of the straight line passing through `(a,b,c)` and parallel to Z-axis?

Options

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

Answer

To find the equation of the straight line passing through the point \((a, b, c)\) and parallel to the Z-axis, we can follow these steps:

### Step 1: Understand the Direction of the Line
A line that is parallel to the Z-axis means it does not change in the X or Y directions, only in the Z direction. Therefore, the direction ratios of the line can be represented as:
- \( \text{Direction ratio along X} = 0 \)
- \( \text{Direction ratio along Y} = 0 \)
- \( \text{Direction ratio along Z} = 1 \)

### Step 2: Use the General Equation of a Line
The general form of the equation of a line in 3D space can be given by:
\[ \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} \]
where \((x_1, y_1, z_1)\) is a point on the line and \((l, m, n)\) are the direction ratios.

### Step 3: Substitute the Known Values
Given the point \((a, b, c)\) and the direction ratios \((0, 0, 1)\):
- \(x_1 = a\), \(y_1 = b\), \(z_1 = c\)
- \(l = 0\), \(m = 0\), \(n = 1\)

Substituting these values into the equation gives:
\[ \frac{x - a}{0} = \frac{y - b}{0} = \frac{z - c}{1} \]

### Step 4: Simplify the Equation
Since the denominators for \(x\) and \(y\) are zero, this indicates that \(x\) and \(y\) are constant:
- \(x = a\)
- \(y = b\)

The equation for \(z\) can be expressed as:
\[ z - c = t \quad \text{(where \(t\) is a parameter)} \]
This implies:
\[ z = c + t \]

### Final Equation of the Line
Combining these results, the equation of the line can be expressed as:
\[ x = a, \quad y = b, \quad z = c + t \]
or in parametric form:
\[ (x, y, z) = (a, b, c + t) \]