Parallel Lines and Their Slopes

Parallel lines are two lines that never intersect each other. They follow the same direction and remain equidistant throughout their length. The slope of a line is a measure of its steepness, and it is defined as the ratio of the change in y-coordinate to the change in x-coordinate between any two points on the line. In other words, the slope of a line is the tangent of the angle that the line makes with the x-axis.

Slope Formula

The formula for the slope of a line passing through two points (x1, y1) and (x2, y2) is given by:

slope = (y2 - y1) / (x2 - x1)

If two lines are parallel, then their slopes are equal. This is because they follow the same direction, and hence their angles with the x-axis are the same. Therefore, the difference in their slopes is zero.

Proof

Suppose we have two lines L1 and L2 with slopes m1 and m2, respectively. Let us assume that the lines are parallel, i.e., they do not intersect each other. Then, for any two points (x1, y1) and (x2, y2) lying on L1 and L2, respectively, we have:

m1 = (y2 - y1) / (x2 - x1)

m2 = (y2 - y1) / (x2 - x1)

Subtracting the second equation from the first, we get:

m1 - m2 = (y2 - y1) / (x2 - x1) - (y2 - y1) / (x2 - x1)

m1 - m2 = 0

Therefore, the difference in slopes of two parallel lines is zero.

Conclusion

In conclusion, we can say that the correct answer to the given question is option 'B' (0), as the difference in slopes of two parallel lines is always zero. This property of parallel lines is very useful in geometry and trigonometry, as it helps in solving problems related to angles and distances.