Problem 5 - Intermediate Algebra

Question

A. Write the standard form of the equation of a circle. B. Write the standard form of the equation of a circle with the center at the origin.

Step-by-Step Solution

Answer

A: \((x - h)^2 + (y - k)^2 = r^2\); B: \(x^2 + y^2 = r^2\).

Step 1: Understanding Standard Form of a Circle Equation

The standard form of a circle's equation is derived from the distance formula. It is expressed as \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle, and \(r\) is the radius.

Step 2: Center of Circle at Origin

When the center of the circle is at the origin, the coordinates \((h, k)\) are \((0, 0)\). This simplifies the equation to \((x - 0)^2 + (y - 0)^2 = r^2\).

Step 3: Simplifying the Equation with Center at Origin

With \((h, k) = (0, 0)\), the equation becomes \(x^2 + y^2 = r^2\). This is the standard circular equation centered at the origin.

Key Concepts

Distance Formula

The distance formula is a vital tool in understanding the geometry of circles, among other shapes. It helps to determine the distance between two points in a plane. The formula is given by:
\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula is derived from the Pythagorean theorem and calculates the straight line distance between the points \((x_1, y_1)\) and \((x_2, y_2)\).

• Used to find the radius of a circle, by measuring from the center to a point on the circle.
• Essential in developing the standard form equation of a circle by substituting the center and a point on the circle.

Understanding how to properly employ the distance formula is foundational to working with circles and further geometric problems.

Circle Equation

The circle equation in its standard form is an elegant representation of all points equidistant from a central point in a plane. The standard form is:
\[(x - h)^2 + (y - k)^2 = r^2\]Here, \((h, k)\) defines the center of the circle and \(r\) is the radius. This equation is powerful in geometry because it encapsulates all necessary information about the circle in a simple format.

• The terms \((x - h)^2\) and \((y - k)^2\) ensure every point on the circle is the same distance, \(r\), from the center.
• The square format is derived from the distance formula, emphasizing the constant radius.

This equation is not just a formula but a tool. It allows you to easily figure out if a point lies on, inside, or outside the circle, simply by calculating and comparing distances.

Center of a Circle

The center of the circle is the referential point from which all points on the edge of the circle are equidistant. Given the standard form equation \((x - h)^2 + (y - k)^2 = r^2\), the center is represented by \((h, k)\).
Understanding the center is crucial because:

• If the center is \((0, 0)\), it greatly simplifies the equation to \(x^2 + y^2 = r^2\), creating a neat symmetry around the origin.
• The role of \(h\) and \(k\) is to shift the circle across the plane without altering its shape, merely changing its position.

Thus, comprehending the concept of the circle’s center not only aids in solving problems involving circles but also in understanding more complex geometrical transformations.