Problem 46 - Intermediate Algebra

Question

Write an equation of the circle with the given center and radius. $$(0,-6) ; \sqrt{2}$$

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + (y + 6)^2 = 2\).

Step 1: Understanding the Circle Equation

The standard equation of a circle in the Cartesian plane with a given center \((h,k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h,k)\) is the center of the circle and \(r\) is the radius.

Step 2: Identify Given Values

From the problem, the center of the circle \((h,k)\) is \((0,-6)\) and the radius \(r\) is \(\sqrt{2}\).

Step 3: Substitute Values into Circle Equation

Substitute \(h = 0\), \(k = -6\), and \(r = \sqrt{2}\) into the circle's equation: \((x - 0)^2 + (y + 6)^2 = (\sqrt{2})^2\).

Step 4: Simplify the Equation

Simplify the equation by calculating \((\sqrt{2})^2\), which is \(2\). The equation becomes \(x^2 + (y + 6)^2 = 2\).

Key Concepts

standard form of a circle

The standard form of a circle's equation is a fundamental concept in geometry, especially when dealing with graphs in a Cartesian plane. The equation is expressed as:\[(x - h)^2 + (y - k)^2 = r^2\]where:

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius of the circle.
• \(x\) and \(y\) are the variables representing any point on the circle moving around the center.

This formula helps define the circle by stating all points \((x,y)\) that are exactly \(r\) units away from the center point \((h,k)\).
This standard form is crucial because it allows us to easily identify and graph the circle when we know the center and radius. It offers a neat summary for describing any circle on a Cartesian coordinate plane.

center of a circle

The center of a circle is the exact middle point from which every point on the circle is equidistant. In the standard form equation of a circle, this center is represented.
It is given by the coordinates \((h,k)\).
For instance, if the circle's center is at \((0, -6)\), the point is located directly on the y-axis in the Cartesian plane, 6 units below the x-axis.The location of the center is crucial because:

• It determines the positioning of the circle within the Cartesian plane.
• It describes how the shape of the circle is centered in relation to the origin \((0,0)\).

Understanding where the center lies helps greatly in graphing the circle accurately. You can imagine starting from the center and tracing the radius outward to form the circle's boundary.

radius of a circle

The radius of a circle is a line segment that connects the center of the circle to any point on the circle's edge.
It is a constant distance from the center that helps define the size of the circle.
In the circle's standard form, the radius is denoted as \(r\).Knowing the value of the radius is vital for:

• Calculating the area and circumference of the circle using respective formulas \(\pi r^2\) and \(2\pi r\).
• Determining the scope or extension the circle covers within the Cartesian plane.

For example, if the radius is given as \(\sqrt{2}\), squaring this value results in 2, which fits into the equation nicely \((x - h)^2 + (y - k)^2 = 2\). This showcases how the radius plays a direct role in shaping the circle.

Cartesian plane

The Cartesian plane is a two-dimensional plane made by the intersection of a horizontal line - the x-axis, and a vertical line - the y-axis. This coordinate system helps in locating points and graphing equations like that of a circle.Key features of the Cartesian plane include:

• Each point on the plane is defined by an ordered pair \((x,y)\).
• The origin, \((0,0)\), where the x-axis and y-axis intersect, serves as a reference point for locating others.

In the context of a circle, the Cartesian plane enables you to map out the position of the circle based on its center \((h,k)\). It provides a clear layout on which we see how the circle's radius extends outward uniformly in all directions from the center, forming a perfectly round shape. Understanding this plane is key to visualizing how geometric shapes and equations manifest in a two-dimensional space.