Question

Find an equation of the line satisfying the conditions. Write your answer in the slope-intercept form. Is perpendicular to the \(x\) -axis and passes through the point \(\left(\pi, \pi^{2}\right)\)

Short answer

The equation of the line perpendicular to the x-axis and passing through the point \((\pi, \pi^2)\) is a vertical line with undefined slope. Therefore, the equation of the line is \(x = \pi\). Note that the slope-intercept form is not applicable for vertical lines.

Step-by-step solution

Determine the slope of the line

Since the line is perpendicular to the x-axis, it must be vertical. Vertical lines have an undefined slope (as the x-coordinates never change). In this case, we don't need to calculate any slope value as vertical lines have no slope (undefined).

Write the equation of the vertical line

Vertical lines have equations of the form x = c, where c is a constant. Since this line passes through the point \((\pi, \pi^2)\), substituting the x coordinate of this point will give us the equation of the line. Thus, the equation of the line is: \[ x = \pi \] Note that the slope-intercept form (y = mx + b) is not applicable to vertical lines, as they have no slope value. So, the final answer is the equation of the vertical line x = π.

Key concepts

Undefined Slope

When we talk about the slope of a line in coordinate geometry, we're referring to how steep the line is. However, not all lines have a slope that we can quantify. Lines that are vertical—parallel to the y-axis—have what is known as an undefined slope. This is because the formula for slope, which is the change in y over the change in x (denoted as \(\frac{\Delta y}{\Delta x}\)), doesn't work for vertical lines. That's because the x-value never changes, no matter how far you move along the line, causing a division by zero situation which is undefined in mathematics.

For example, consider the equation \( x = \pi \) in our exercise. Here, no matter what y-value you choose, x remains constant at \( \pi \). This absence of x-change means the slope cannot be calculated and is thus undefined. Remember, anytime a line is straight up and down, you're dealing with a line with an undefined slope.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that you can easily see the slope and the y-intercept, the point where the line crosses the y-axis. This form is written as \( y = mx + b \), with \( m \) representing the slope and \( b \) representing the y-intercept. This form is convenient as it allows for quick graphing of the line and understanding of its behavior.

Most lines can be expressed in this form, but as we saw in the exercise, vertical lines are an exception. Since vertical lines have an undefined slope, we cannot use the slope-intercept form for them. Instead, their equation is given simply by the constant x-value they pass through, which is a clear deviation from the typical y equals expression.

Perpendicular Lines

In coordinate geometry, when two lines are perpendicular to each other, they meet at a right angle (90 degrees). An important property to remember is that the slopes of two perpendicular lines are negative reciprocals of each other, given neither is vertical. This means if one line has a slope of \( m \), the line perpendicular to it will have a slope of \( -\frac{1}{m} \) if \( m \) is not zero or undefined.

In our exercise, we don't calculate a reciprocal since the line is perpendicular to the x-axis and thus has an undefined slope. However, when dealing with non-vertical perpendicular lines, calculating the slope of one allows us to find the slope of the other using this negative reciprocal relationship.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful branch of mathematics that allows us to describe and analyze the relationships of geometric figures using an x-y coordinate system. Concepts like slope, slope-intercept form, and perpendicularity are fundamental in understanding complex geometric shapes and their behaviors in a plane.

Applications of coordinate geometry include finding distances between points, determining the areas and perimeters of shapes, and solving real-world problems involving maps, blueprints, and even navigation. The beauty of using a coordinate system lies in its ability to transform visual information into algebraic equations, making it easier to manipulate and interpret geometric data. Understanding the basic principles, like those illustrated in our textbook exercise, provides the tools necessary for delving into more advanced topics within this subject.