Final answer:
The line passing through the points (p, a) and (p, -a) is a vertical line with an undefined slope, the equation x = p, no y-intercept, and a slope of 0 for a perpendicular line.
Step-by-step explanation:
The line that passes through the points (p, a) and (p, -a), with p and a being real numbers and p ≠ 0, has certain characteristics that can be derived using the concepts of slope and y-intercept for linear equations.
1. Slope of the line
The slope of a line is calculated by the change in y divided by the change in x (rise over run). Here, the change in y is a - (-a) = 2a, and the change in x is p - p = 0. Since dividing by zero is undefined, the slope of this line is undefined. This means we are dealing with a vertical line.
2. Equation of the line
The equation of a vertical line is x = c, where c is the x-coordinate that the line passes through. In this case, the equation of our line is x = p.
3. y-intercept
Since the line is vertical, it does not intersect the y-axis (unless p equals 0, which it does not in this case); therefore, it has no y-intercept.
4. Slope of a line perpendicular to the given line
A line perpendicular to a vertical line is a horizontal line, which has a slope of 0. This is because there is no vertical change as we move along the line, so the rise over run calculation equals 0 divided by any non-zero number, which gives a result of 0.