Question

A line passes through the points (p, a) and (p, -a) where p and a are real numbers andp = 0. Describe each of the
following. Explain your reasoning.
ht.
slope of the line
2. equation of the line
3. y-intercept
4. slope of a line perpendicular to the given line

Answer 1

Answers:

1. slope = undefined
2. equation is x = p
3. there is no y intercept
4. slope of perpendicular line is 0

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Explanation:

1)

If we were to apply the slope formula for the points (p,a) and (p,-a), then we will have x2-x1 = p-p = 0 in the denominator. This directly leads to a division by zero error so the slope is undefined. This applies to any vertical line.

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2)

The equation of the line through (p,a) and (p,-a) is x = p since it describes every point on this line has an x coordinate of p. The y coordinate doesn't matter which means the y variable is left out of the equation.

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3)

If $p \ne 0$, then (p,a) and (p,-a) forms a vertical line parallel to the y axis. This vertical line never crosses the y axis. There's no y intercept in this case.

If p = 0, then the vertical line through (p,a) and (p,-a) will pass through infinitely many points on the y axis and have infinitely many y intercepts. When your teacher asks for the y intercept, they are only asking for one value. Like the previous paragraph, I consider the outcome here to be "no y intercept".

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4)

The slope we found back in problem 1 had 0 in the denominator. If we applied the negative reciprocal operation, then 0 goes to the numerator and makes the perpendicular slope to be 0. Any horizontal line has a slope of 0. It's to indicate there's no vertical change (aka rise).

In short,

• vertical line: slope is undefined
• horizontal line: slope is zero