Distance between `P (x_(1), y_(1))` and `Q (x_(2), y_(2))` when PQ is parallel to y - axis is

Options

A. `x_(1)-x_(2)`
B. `|x_(1)-x_(2)|`
C. `y_(1)-y_(2)`
D. `|y_(1)-y_(2)|`

Answer

To find the distance between the points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) when the line segment \( PQ \) is parallel to the y-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Condition**: Since the line segment \( PQ \) is parallel to the y-axis, it means that both points \( P \) and \( Q \) have the same x-coordinate. Therefore, we can say:
\[ x_1 = x_2 \]

2. **Distance Formula**: The distance \( d \) between two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) in a Cartesian coordinate system is given by the formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

3. **Substituting the x-coordinates**: Since \( x_1 = x_2 \), we can substitute \( x_2 \) with \( x_1 \) in the distance formula:
\[ d = \sqrt{(x_1 - x_1)^2 + (y_2 - y_1)^2} \]

4. **Simplifying the Expression**: The term \( (x_1 - x_1)^2 \) becomes 0:
\[ d = \sqrt{0 + (y_2 - y_1)^2} \]
\[ d = \sqrt{(y_2 - y_1)^2} \]

5. **Taking the Square Root**: The square root of a squared term is the absolute value:
\[ d = |y_2 - y_1| \]

### Final Answer:
Thus, the distance between the points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \) when \( PQ \) is parallel to the y-axis is:
\[ d = |y_2 - y_1| \]