Chapter 1.3, Problem 77E

a .

To determine

To find: the equation of line that passes through the point $\left(-2,4\right)$ and is parallel to the line $y+5=0$ .

Answer to Problem 77E

  $y=4$

Explanation of Solution

Concept Used:

Slope intercept form of equation of line is $y=mx+b$ , where m is the slope of the line and $\left(0,b\right)$ is the y -intercept of the line.

Point slope form of equation of line is $\left(y-y_1\right)=m\left(x-x_1\right)$ , where m is the slope of the line and line passes through the point $\left(x_1,y_1\right)$ .

Consider,

  $\begin{array}{c}y+5=0\\ y=-5\end{array}$

The lines of the form $y=b$ are horizontal lines and have slope $m=0$ .

So, the slope of this line is $m=0$ .

Now, two lines are parallel to each other only if they have the same slope.

So, the required line have slope $m=0$ and it is given that is passes through the point $\left(-2,4\right)$ .

Then by point slope form, the equation of line is,

  $\begin{array}{c}\left(y-y_1\right)=m\left(x-x_1\right)\\ y-4=0\left(x-\left(-2\right)\right)\\ y-4=0\\ y=4\end{array}$

Conclusion:

The equation of line that passes through the point $\left(-2,4\right)$ and is parallel to the line $y+5=0$ is $y=4$ .

b .

To determine

To find: the equation of line that passes through the point $\left(-2,4\right)$ and is perpendicular to the line $y+5=0$ .

Answer to Problem 77E

  $x=-2$

Explanation of Solution

Concept Used:

Slope intercept form of equation of line is $y=mx+b$ , where m is the slope of the line and $\left(0,b\right)$ is the y -intercept of the line.

Point slope form of equation of line is $\left(y-y_1\right)=m\left(x-x_1\right)$ , where m is the slope of the line and line passes through the point $\left(x_1,y_1\right)$ .

Consider,

  $\begin{array}{c}y+5=0\\ y=-5\end{array}$

The lines of the form $y=b$ are horizontal lines and have slope $m=0$ .

So, the slope of this line is $m=0$ .

Now, two lines are perpendicular to each other only if their slopes are negative reciprocals of each other.

So, the required line have slope $m=-\frac{1}{0}$ that is it have undefined slope. This implies that the line is a vertical line of the form $x=a$

Also it is given that is passes through the point $\left(-2,4\right)$ .

Substitute $\left(-2,4\right)$ in the equation $x=a$ to find value of a ,

  $\begin{array}{c}x=a\\ -2=a\end{array}$

Thus, the equation of required line is,

  $x=-2$

Conclusion:

The equation of line that passes through the point $\left(-2,4\right)$ and is perpendicular to the line $y+5=0$ is $x=-2$ .