Chapter 0, Problem 68EP

(a)

To determine

To calculate : The slope of line $L$ .

Answer to Problem 68EP

The slope of line $L$ is $-1$ .

Explanation of Solution

Given information :

The point $P\left(-2,1\right)$ and the line $L:x+y=2$ .

Formula used:

The slope of line $y=mx+c$ is $m$ .

Calculation:

The equation can be re-written as

  $y=-x+2$

Comparing this equation with $y=mx+c$

We get the slope of the line as:

  $m=-1$ .

(b)

To determine

To calculate : The equation for the line through $P$ and parallel to $L$ .

Answer to Problem 68EP

The equation for the line through $P$ and parallel to $L$ is $y=-x-1$ .

Explanation of Solution

Given information :

The point $P\left(-2,1\right)$ and the line $L:x+y=2$ .

Formula used:

The slope of the parallel lines is same.

Calculation:

Now for a line to be parallel to $L:x+y=2$ at the given point, the slopes of the two lines are equal.

So slope of the required line is $-1$ .

Let the equation of the line parallel to the given line be

  $y=-x+c$ …..(2)

As it passes through $P\left(-2,1\right)$ ,

Then

  $\begin{array}{l}\text{I}=\left(-1\right)\left(-2\right)+c\\ \Rightarrow c=-1\end{array}$

Therefore, the equation of the required line is $y=-x-1$ .

(c)

To determine

To calculate : The equation for the line through $P$ and perpendicular to $L$ .

Answer to Problem 68EP

The equation for the line through $P$ and perpendicular to $L$ is $y=x+3$ .

Explanation of Solution

Given information :

The point $P\left(-2,1\right)$ and the line $L:x+y=2$ .

Formula used:

The product of slope of perpendicular lines is $-1$ .

Calculation:

Now for a line to be perpendicular to the given line at the given point, the product of the slopes of the two lines should be equal to $-1$ .

So slope of the required line is 1.

Let the equation of the line perpendicular to the given line be

  $y=x+c$ …..(2)

As it passes through $P\left(-2,1\right)$ ,

Then

  $\begin{array}{l}\text{1}=\left(1\right)\left(-2\right)+c\\ \Rightarrow c=3\end{array}$

Therefore, the equation of the required line is $y=x+3$ .

(d)

To determine

To calculate : The $x$ intercept of $L$ .

Answer to Problem 68EP

The $x$ intercept of $L$ is $2$ .

Explanation of Solution

Given information :

The point $P\left(-2,1\right)$ and the line $L:x+y=2$ .

Calculation:

To find $x$ -intercept of $L$ ,

Put $y=0$ in $L:x+y=2$ . So,

  $\begin{array}{c}x+0=2\\ x=2\end{array}$

Therefore, $x$ -intercept of $L$ is $2$ .