Chapter 10.1, Problem 94E

a.

To determine

Find the distance d between the point and line as a function of $m$ .

Answer to Problem 94E

  $d=\frac{\left|3m+3\right|}{\sqrt{1+m^2}}$

Explanation of Solution

Calculation:

Consider a line with slope $m$ and $y$ -intercept $c$ ,is given by, $y=mx+c$

Since, $y$ -intercept $\left(0,4\right)$

  $c=4$

The equation of the line is, $y=mx+4$

The distance between the point $\left(x_1,y_1\right)$ and the line $Ax+By+C=0$

  $d=\frac{\left|Ax+By+C\right|}{\sqrt{A^2+B^2}}$

Now the distance between line $y=mx+4$ and point $\left(3,1\right)$ ,

  $\begin{array}{l}d=\frac{\left|m{x}_1-y_1+4\right|}{\sqrt{1^2+m^2}}\\ =\frac{\left|3m-1+4\right|}{\sqrt{1+m^2}}\\ d=\frac{\left|3m+3\right|}{\sqrt{1+m^2}}\end{array}$

Hence, the distance $d$ between the point $\left(3,1\right)$ and the line as a function of $m$ is $d=\frac{\left|3m+3\right|}{\sqrt{1+m^2}}$

b.

To determine

Graph the function $d=\frac{\left|3m-1+4\right|}{\sqrt{1+m^2}}$

Answer to Problem 94E

  

Explanation of Solution

Given information:

Consider a line with slope $m$ and $y$ -intercept $\left(0,4\right)$ .Graph the function in part $\left(a\right)$ .

Calculation:

With the help of maple plot function $d=\frac{\left|3m-1+4\right|}{\sqrt{1+m^2}}$

  

Hence, the function plotted.

c.

To determine

Find the slope that yields the maximum distance between the point and the line.

Answer to Problem 94E

  $m=1$

Explanation of Solution

Calculation:

To find the maximum distance between the point and the line let differentiate with respect to slope $m$ .

  $\begin{array}{l}\frac{dd}{dm}=0\\ \frac{d}{dm}=\frac{\left|3\left(m+1\right)\right|}{\sqrt{1+m^2}}=0\\ \frac{d}{dm}=\frac{\left|\left(m+1\right)\right|}{\sqrt{1+m^2}}=0\\ \frac{\sqrt{1+m^2}-\left(m+1\right)\frac{1}{2}\frac{2m}{\sqrt{1+m^2}}}{\sqrt{1+m^2}}=0\\ \frac{\left(1+m^2\right)-\left(1+m\right)m}{2{\left(1+m^2\right)}^{3/2}}=0\\ m=1\end{array}$

Hence, the slope that yields the maximum distance between the point $\left(3,1\right)$ and the line is $m=1$

d.

To determine

Find the slope of the line that yields a distance of $0$ .

Answer to Problem 94E

  $m=-1$

Explanation of Solution

Calculation:

To find the slope of the line that yields a distance of $0$ ,

When $d=0$

  $\begin{array}{l}\frac{\left|3\left(m+1\right)\right|}{\sqrt{1+m^2}}=0\\ m=-1\end{array}$

Hence, the slope of the line that yields a distance of $0$ is $m=-1$

e.

To determine

Find the asymptote of the graph in part $\left(b\right)$ .

Answer to Problem 94E

  $d=3$

Explanation of Solution

Calculation:

The horizontal asymptote is as shown.

  $\begin{array}{l}\underset{m\to \infty }{\lim d}=\underset{m\to \infty }{\lim }\left(\frac{\left|3\left(m+1\right)\right|}{\sqrt{1+m^2}}\right)\\ =\underset{m\to \infty }{\lim }\left(\frac{\left|3\left(1+\frac{1}{m}\right)\right|}{\sqrt{1+\frac{1}{m^2}}}\right)\\ =3\end{array}$

Hence, $d=3$ is horizontal asymptote.