Question

A line passes through the points (p, a) and (p, -a) where p and a are real numbers, and p = 0. Describe each of the following:

What is the slope of the line?
a) 0
b) Undefined
c) 1
d) -1

What is the equation of the line?
a) x = p
b) y = a
c) y = -a
d) x = a

What is the y-intercept of the line?
a) (0, a)
b) (0, -a)
c) (a, 0)
d) (-a, 0)

What is the slope of a line perpendicular to the given line?
a) 0
b) Undefined
c) 1
d) -1

Answer 1

The slope of the vertical line passing through (p, a) and (p, -a) with p = 0 is undefined. The equation of this line is x = 0. The line contains all points on the y-axis, so it does not have a single y-intercept, but a slope of 0 would describe a line perpendicular to it.

Step-by-step explanation:

The line in question passes through two points where the x-coordinate remains the same, indicating that this is a vertical line. The slope of a vertical line is undefined, because the change in y over the change in x (rise over run) involves a division by zero, which is not defined in mathematics. Undefined is the answer for the slope of this line.

The equation of a vertical line is expressed as x equals a constant value, which in this case is x = p, where p is specified as 0. Therefore, the correct equation is x = 0.

The y-intercept occurs where the line crosses the y-axis, which is at the point (0, y). Since our line is x = 0, it intersects the y-axis along every point where x is 0; thus, it does not have a single y-intercept point but instead contains all points where x is 0. However, in the given options, none is technically correct as the line itself is on the y-axis.

Finally, a line perpendicular to a vertical line is a horizontal line, which has a slope of 0, because there is no rise over the horizontal run.