Chapter 0.1, Problem 31E

(a)

To determine

To write the equation of the line passing through the point $P\left(-2,4\right)$ and parallel to the line $L:x=5$ .

Answer to Problem 31E

The line parallel to $L:x=5$ and passing through the point $P\left(-2,4\right)$ will be $x=-2$ .

Explanation of Solution

Given:

Given a point $P\left(-2,4\right)$ and a line $L:x=5$ .

Calculation:

Observe that the line $L:x=5$ is a vertical one.

Thus, the lines parallel to $L:x=5$ are also vertical lines.

Now, the vertical line passing through the point $P\left(-2,4\right)$ is $x=-2$ .

Thus, the line parallel to $L:x=5$ and passing through the point $P\left(-2,4\right)$ will be $x=-2$ .

Conclusion:

The line parallel to $L:x=5$ and passing through the point $P\left(-2,4\right)$ will be $x=-2$ .

(b)

To determine

To write the equation of the line passing through the point $P\left(-2,4\right)$ and perpendicular to the line $L:x=5$ .

Answer to Problem 31E

The equation of the line perpendicular to $L:x=5$ and passing through the point $P\left(-2,4\right)$ is $y=4$ .

Explanation of Solution

Given:

Given a point $P\left(-2,4\right)$ and a line $L:x=5$ .

Calculation:

Since the line $L:x=5$ is a vertical one, lines perpendicular to $L:x=5$ are all horizontal ones having the general form $y=a$ .

Now, it is required to find the equation of the line passing through the point $P\left(-2,4\right)$ .

Now, the equation of the horizontal line passing through the point $P\left(-2,4\right)$ is $y=4$ .

Thus, the equation of the line perpendicular to $L:x=5$ and passing through the point $P\left(-2,4\right)$ is $y=4$ .

Conclusion:

The equation of the line perpendicular to $L:x=5$ and passing through the point $P\left(-2,4\right)$ is $y=4$ .