Question

In Exercise \(9-12,\) write an equation for (a) the vertical line and (b) the horizontal line through the point \(P .\) $$P(0,-\sqrt{2})$$

Short answer

The equation of the vertical line is \(x = 0\) and the equation of the horizontal line is \(y = -\sqrt{2}\).

Step-by-step solution

Write Equation for Vertical Line

To write the equation for the vertical line through the given point, we use the \(x\)-coordinate of the given point in the overall equation \(x = a\). Here, \(a\) is the \(x\)-coordinate of P which is \(0\), so the equation of the vertical line is \(x = 0\).

Write Equation for Horizontal Line

Now, to write the equation for the horizontal line through the given point, we use the \(y\)-coordinate of the point in the equation \(y = b\). Here, \(b\) is the \(y\)-coordinate of P which is \(-\sqrt{2}\), so the equation of the horizontal line is \(y = -\sqrt{2}\).

Key concepts

Vertical Line Equation

A vertical line in a coordinate plane is one that runs up and down, parallel to the y-axis. Its unique characteristic is that all points along the line have the same x-coordinate. To express this in an equation, we only need this x-coordinate, resulting in an equation of the form
\( x = a \).
If you need to find the equation of a vertical line through a particular point, you use the x-coordinate of that point for 'a'. For instance, if a line passes through the point \( (0,-\sqrt{2}) \) as in our example, the vertical line equation is simply \( x = 0 \), because the x-coordinate of the point is 0. This is an algebraic representation of a vertical line in the coordinate plane and it tells us that no matter what y-value you have, the x-value will always be 0.

Horizontal Line Equation

In contrast to a vertical line, a horizontal line runs left to right, parallel to the x-axis. For every point on a horizontal line, the y-coordinate is the same, leading to an equation of the form \( y = b \).
When you need to write the equation for a horizontal line that goes through a specific point, you take the y-coordinate of that point and substitute it in for 'b'. For the example given, with the point \( (0,-\sqrt{2}) \), the y-coordinate is \( -\sqrt{2} \), so the equation of the horizontal line would be \( y = -\sqrt{2} \). This shows a direct relationship, indicating that regardless of the x-value, the y-value will always be this constant, which is \( -\sqrt{2} \) in our case.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system allows for the algebraic representation of geometric shapes and can facilitate the process of solving geometric problems.
In coordinate geometry, points are described by coordinates, typically \( (x, y) \) on the Cartesian plane. Lines, on the other hand, are often described by equations that relate these coordinates to each other. As demonstrated in our example, the point \( (0,-\sqrt{2}) \) is used to define the positions of a vertical and horizontal line through algebraic equations. By combining algebra and geometry in this way, coordinate geometry enables us to analyze and solve problems concerning the shapes and positions of geometric figures in a plane.

Algebraic Representation

The algebraic representation in mathematics is a way of using algebraic expressions and equations to describe mathematical objects and relationships between them.
In the context of finding equations of lines, algebraic representation offers a straightforward approach to describe lines in terms of equations. The equations \( x = a \) and \( y = b \) are examples of how the location and orientation of vertical and horizontal lines, respectively, are represented algebraically.
Moreover, using algebra to describe a line allows for efficient communication and calculations in mathematics. By understanding that a vertical line has a constant x-value, and a horizontal line has a constant y-value, you can quickly determine how these lines appear and behave on a coordinate plane. Algebra serves as the language that bridges numerical data with geometric figures, enhancing our ability to comprehend and work with the spatial relationships in mathematics.