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 
                               $\displaystyle{ m=\frac{d-b}{c-a},$

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

                               $\displaystyle{ m=\frac{-a-a}{p-p}=\frac{-2a}{0}.$

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.