Question

A line with slope \(m\) passes through the point (0,-2) . (a) Write the distance \(d\) between the line and the point (4,2) as a function of \(m\). (b) Use a graphing utility to graph the equation in part (a). (c) Find \(\lim _{m \rightarrow \infty} d(m)\) and \(\lim _{m \rightarrow-\infty} d(m)\). Interpret the results geometrically.

Short answer

The distance function is \(d(m)=\frac{|4 - 4m|}{\sqrt{m^2 + 1}}\), which must be graphed using a graphing utility. The limits of \(d(m)\) as \(m\) approaches infinity and negative infinity are -1 and 1 respectively.

Step-by-step solution

Calculate the distance d

The slope intercept form of a straight line is \(y = mx + c\). Given the slope \(m\) and knowing that the line passes through the point (0,-2), the equation of the line is \(y=mx-2\). The distance \(d\) between the point (4,2) and the line can be calculated using the distance formula \(d(m)=\frac{|y_1 - mx_1 + c|}{\sqrt{m^2 + 1}}\), which results in \(d(m)=\frac{|2 - m*4 + 2|}{\sqrt{m^2 + 1}} = \frac{|4 - 4m|}{\sqrt{m^2 + 1}}\)

Graph d(m)

To graph the function \(d(m)=\frac{|4 - 4m|}{\sqrt{m^2 + 1}}\), you can use a graphing utility and plot the function for a range of \(m\) values. Note that because of the absolute value | |, this is a mirrored graph around \(m=1\).

Calculate limits

To calculate the limit of \(d(m)\) as \(m\) tends towards infinity and negative infinity, use the limit properties and rules. The function \(d(m)=\frac{|4 - 4m|}{\sqrt{m^2 + 1}}\) is a quotient of functions, so the limit can be found using the quotient rule of limits. Calculating \(\lim _{m \rightarrow \infty} d(m) = \lim _{m \rightarrow \infty} \frac{|4 - 4m|}{\sqrt{m^2 + 1}} = \frac{-\infty}{\infty}= -1 \). Calculating \(\lim _{m \rightarrow -\infty} d(m) = \lim _{m \rightarrow -\infty} \frac{|4 - 4m|}{\sqrt{m^2 + 1}} = \frac{\infty}{\infty}=1\). The geometric interpretation of these limits is that as the slope of the line gets steeper, in both the positive and negative direction, the distance to the point (4,2) gets closer to a constant value.