Problem 15 - Thomas Calculus

Question

In Exercises 13–16, find an equation for (a) the vertical line and (b) the horizontal line through the given point. $$ (0,-\sqrt{2}) $$

Step-by-Step Solution

Answer

Vertical line: \(x = 0\), Horizontal line: \(y = -\sqrt{2}\).

Step 1: Identify the coordinates of the point

The given point is \((0, -\sqrt{2})\). Here, the x-coordinate is 0 and the y-coordinate is \(-\sqrt{2}\).

Step 2: Write the equation for the vertical line

A vertical line has a constant x-coordinate for all points on the line. Therefore, the equation for the vertical line passing through \((0, -\sqrt{2})\) is simply \(x = 0\).

Step 3: Write the equation for the horizontal line

A horizontal line has a constant y-coordinate for all points on the line. Thus, the equation for the horizontal line passing through \((0, -\sqrt{2})\) is \(y = -\sqrt{2}\).

Key Concepts

Vertical Line

A vertical line is a straight line that goes up and down, parallel to the y-axis on a coordinate plane. In mathematics, a vertical line is represented by an equation where the x-coordinate is always the same for any point on the line.

For example, consider a line passing through the point \(0, -\sqrt{2}\). Here, the x-value is consistently zero, making the equation of the vertical line \(x = 0\).

Key features of a vertical line include:

• It cannot be expressed as \(y = mx + b\).
• Its slope is undefined because there is no horizontal change.
• It shows an infinite number of y-values for a single x-value along the line.

These properties make vertical lines unique, as they distinctively refuse to "run," giving only a "rise." This means vertical lines do not have a slope in traditional sense, contrasting with most other linear equations.

Horizontal Line

Horizontal lines are lines that stretch from left to right across the coordinate plane, parallel to the x-axis. They demonstrate a consistent y-coordinate for every point on the line.

An equation for a horizontal line is always in the format of \(y = c\), where \(c\) is a constant reflecting the y-coordinate for all points on the line. Consider, for example, a horizontal line through the point \(0, -\sqrt{2}\). The equation thus becomes \(y = -\sqrt{2}\), because the y-value remains the same across the line.

Important aspects of horizontal lines:

• They have a slope of zero, since there is no vertical change.
• The line runs endlessly in the horizontal direction, but y stays constant.
• They can be expressed as \(y = ext{constant}\), which highlights their stability in y-value.

Horizontal lines are particularly simple because they maintain this same level without ever dipping or climbing, making them very predictable.

Coordinate Geometry

Coordinate geometry, sometimes known as analytic geometry, is a branch of geometry where points, lines, and other figures are defined using coordinates and algebraic formulas. It serves as a link between algebra and geometry, bridging the gap to solve geometrical problems through numerical methods.

In a coordinate plane, each point is defined by a pair of numbers, usually referred to as \( (x, y) \). These numbers correspond to the point's position relative to two intersecting perpendicular lines (axes).

Coordinate geometry provides the basis for:

• Determining equations of various geometric shapes, such as lines, circles, and parabolas.
• Calculating distances between points using the distance formula.
• Finding the midpoint of a segment with the midpoint formula.

It allows us to translate geometric diagrams into algebraic equations, offering a versatile way to analyze linear equations, tackle complex figures, and perform transformations and resizing, which are key skills in various branches of mathematics and applied fields.