Problem 26 - 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 is \((x - h)^2 + (y - k)^2 = r^2\), representing a circle with center \(C(h, k)\) and radius \(r\).

Step 1: Understand the Problem

We are tasked with finding a formula that represents all points \(P(x, y)\) that are at a distance \(r\) from a fixed point \(C(h, k)\). This is the definition of a circle, where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Use the Distance Formula

To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 3: Apply Distance to Circle Context

Since we want to find the set of points \(P(x, y)\) that are distance \(r\) from \(C(h, k)\), we set up the equation using the distance formula: \(\sqrt{(x - h)^2 + (y - k)^2} = r\).

Step 4: Square Both Sides

To eliminate the square root, square both sides of the equation: \((x - h)^2 + (y - k)^2 = r^2\).

Step 5: Identify the Formula

The equation \((x - h)^2 + (y - k)^2 = r^2\) is the formula for a circle centered at \((h, k)\) with radius \(r\).

Step 6: Describe the Set of Points

The set of all points \((x, y)\) that satisfy \((x - h)^2 + (y - k)^2 = r^2\) forms a circle with center \(C(h, k)\) and radius \(r\).

Key Concepts

Distance Formula

The distance formula is a pivotal mathematical tool that helps us determine the distance between two points in a plane. If you have two points, \(P_1(x_1, y_1)\) and \(P_2(x_2, y_2)\), the distance \(d\) between them is calculated using the formula:

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

This formula is derived from the Pythagorean theorem, making use of the horizontal and vertical distances between the two points.
This is crucial because it lays the groundwork for understanding the equation of a circle in a coordinate plane.
In the context of a circle, our two points are:

• a fixed point \(C(h, k)\) (the center of the circle),
• and a point on the circle \(P(x, y)\).

This is essentially why the distance formula is instrumental in creating the equation that describes a circle.

Center of a Circle

The center of a circle is a critical concept when studying circles. In a coordinate plane, it is usually denoted by \(C(h, k)\).
The center controls where the circle is located. Think of it as the circle's anchor point.
Understanding the center is essential because any changes to \(h\) or \(k\) will shift the entire circle horizontally or vertically across the plane.

• Positioned at \(C(h, k)\), it serves as the reference point from which every point on the circle is equidistant.
• This distance is known as the radius.

By knowing the center, you can pinpoint where the circle sits on the grid. When plotting circles, always start by marking the center and then using the radius to draw the perimeter.

Radius of a Circle

The radius of a circle, usually represented as \(r\), is the distance from the center to any point on the circle. In a circle's equation, this distance is a constant and is fundamental to defining the circle's size.
When you write the equation \((x - h)^2 + (y - k)^2 = r^2\), the radius is the \(r\) part:

• \(r\) tells you how far out the circle extends from the center point \(C(h, k)\).
• The larger the \(r\), the bigger the circle.

Understanding the radius is crucial because it directly affects the area and circumference of the circle.
Every point \(P(x, y)\) on the circle maintains this constant distance \(r\) from the center, creating the circular shape we are familiar with.