Question

A line passes through the points (p, a) and (p, –a) where p and a are real numbers. If p=0, what is the y-intercept? Explain your reasoning.

Answer 1

The slope of the line passing through the points A(a, b), and B(c, d), is given by

`\displaystyle{ m=(d-b)/(c-a),`

thus, the slope of the line through points (p, a) and (p, –a) is


`\displaystyle{ m=(-a-a)/(p-p)=(-2a)/(0).`

When applying the slope formula results in division by zero, this means that the line is a vertical line.

This could have also been noticed directly, since the line contains 2 points with the same x-coordinate.

The equation of this line is x=p, since the x-coordinate is always p, no matter what y is.

If p=0, then the equation of the line is x=0, which is the y-axis itself. In this case, the y-intercept is the whole y-axis.

Answer: The whole y-axis.