Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Find a formula that states that \(P(x, y)\) is a distance \(r>0\) from a fixed point \(C(h, k)\). Describe the set of all such points.

Step-by-Step Solution

Answer

The formula \((x - h)^2 + (y - k)^2 = r^2\) describes a circle with center \(C(h, k)\) and radius \(r\).

Step 1: Understand the Problem

We need to find a formula that describes all points \((x, y)\) that are at a distance \(r\) from a fixed point \(C(h, k)\). This set of points forms a geometric figure.

Step 2: Use the Distance Formula

The distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Here, \((x_1, y_1)\) is \((h, k)\) and \((x_2, y_2)\) is \((x, y)\).

Step 3: Set Up the Equation

According to the problem, the distance \(d\) is \(r\). Thus, we set up the equation: \[\sqrt{(x - h)^2 + (y - k)^2} = r\]

Step 4: Eliminate the Square Root

Square both sides of the equation to remove the square root, yielding: \((x - h)^2 + (y - k)^2 = r^2\).

Step 5: Interpret the Result

The equation \((x - h)^2 + (y - k)^2 = r^2\) represents a circle centered at point \(C(h, k)\) with radius \(r\). All points \((x, y)\) that satisfy this equation are on the circumference of the circle.

Key Concepts

Distance Formula

Understanding the distance formula is crucial in coordinate geometry as it helps determine the distance between two points on a coordinate plane. To find this, we use the formula:

• \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

This formula calculates the Euclidean distance, which is like measuring the straight-line distance between two points—just as you would use a ruler. In the case of our exercise, the points are

• \((x_1, y_1) = (h, k)\), the fixed center of the circle
• \((x_2, y_2) = (x, y)\), any point on the circle's circumference

By applying the distance formula, you verify that all points \((x, y)\) maintain a consistent distance \(r\) from the center point \((h, k)\), adhering to the circle's definition.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides a powerful toolset for exploring geometric concepts through algebra. This branch of mathematics uses coordinates on the Cartesian plane to represent geometrical figures, like lines, curves, and here, circles.
By transforming geometric shapes into equations, coordinate geometry allows us to manipulate and analyze these shapes algebraically.
In our context, a circle is a set of all points equidistant from a center point \((h, k)\).
Representing a circle algebraically allows for:

• Easy determination of whether a point lies on the circle.
• The ability to find the intersection points with other geometric figures.
• Calculation of properties like diameter and area.

This approach creates a bridge between algebra and geometry, making complex concepts more accessible and applicable.

Equation of a Circle

The equation \((x - h)^2 + (y - k)^2 = r^2\) is a standard form for representing circles. Understanding each component of this equation:

• \((h, k)\) represents the circle's center
• \(r\) is the radius, which defines the distance from the center to the circle's edge

This equation fundamentally implies that every point \((x, y)\) meeting this requirement lies perfectly on the circle. To construct this equation:

• Use the distance formula to equate distance, \(r\), between the center \((h, k)\) and any point \((x, y)\)
• Square both sides of the equation to remove the square root, simplifying computation

By solving this equation, one can determine points on the circle, proving it as a tangible shape defined by predictable rules.