Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Find all points on the \(x\)-axis that are a distance 5 from \(P(-2,4)\).

Step-by-Step Solution

Answer

The points on the \(x\)-axis are \((1, 0)\) and \((-5, 0)\).

Step 1: Understanding the Problem

We need to find points on the \(x\)-axis that are a distance of 5 units from the given point \(P(-2, 4)\). Since these points lie on the \(x\)-axis, their coordinates will be in the form \((x, 0)\).

Step 2: Distance Formula Setup

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). For a point \((x, 0)\) on the \(x\)-axis and \(P(-2, 4)\), the distance is 5. So, we set up the equation: \[\sqrt{(x - (-2))^2 + (0 - 4)^2} = 5\].

Step 3: Simplifying the Equation

Simplify the distance equation: \[\sqrt{(x + 2)^2 + 16} = 5\].To remove the square root, square both sides:\[(x + 2)^2 + 16 = 25\].

Step 4: Solve for \(x\)

Subtract 16 from both sides:\[(x + 2)^2 = 9\].Take the square root of both sides:\[x + 2 = \pm 3\].Solve for \(x\) to find:\[x = 1\] and \[x = -5\].

Step 5: Verify Solution

Check the points \((1, 0)\) and \((-5, 0)\) by substituting into the distance formula with \(P(-2, 4)\) to ensure the distance is indeed 5 units. Calculating confirms both points satisfy the conditions.

Key Concepts

Distance Formula

The distance formula is crucial for calculating the distance between two points in a coordinate plane. It is derived from the Pythagorean theorem and helps us find how far apart two points are. Given two points,

• \((x_1, y_1)\)
• \((x_2, y_2)\)

we use the formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] where \(d\) represents the distance. It's simple yet powerful in connecting algebra with geometry, allowing us to calculate distances quickly and accurately.

To use this formula, plug in the coordinates of the two points you are working with, then follow the order of operations. Remember to square the differences, sum them up, and finally take the square root to find the distance. This formula is a go-to tool for many geometry problems.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra with geometry using coordinate systems. This graph-based methodology allows us to represent geometric figures like points, lines, and shapes using coordinates, typically in a 2D plane.

In coordinate geometry, each point is defined by an ordered pair,

• \((x, y)\) in a 2D space

where \(x\) and \(y\) are numbers representing the point's position along the horizontal and vertical axes, respectively. This framework helps in understanding geometric concepts via numerical expressions.

Coordinate geometry simplifies the process of calculating distances, angles, intersections, and many other aspects of shapes and lines. In the given exercise, using coordinate geometry allows us to visualize and compute the points on the \(x\)-axis a certain distance away from another point \(P(-2, 4)\), effectively marrying spatial reasoning with mathematical equations.

X-Axis Symmetry

X-axis symmetry refers to the property of a shape or graph that is mirrored across the x-axis. In other words, if flipping an object or graph over the x-axis results in the same shape, it has x-axis symmetry. This concept is crucial when dealing with graphs of functions or geometric shapes.

In the context of the exercise, x-axis symmetry implies that points on the x-axis maintain certain properties when reflected over the x-axis. For example, a point on the x-axis,

• \((x, 0)\)

might mirror another point above or below the x-axis at equal distances, maintaining symmetry concerning the x-axis.

Recognizing symmetry can help simplify solutions to geometry problems. It often reduces complex problems to more manageable tasks, because it exploits balance and repetition features. This acknowledgment helps us focus only on relevant coordinates, knowing that other parts are naturally aligned or mirrored.