Problem 61 - College Algebra with Modeling and Visualization

Question

Find the center and radius of the circle. $$ x^{2}+y^{2}=7 $$

Step-by-Step Solution

Answer

The center is (0, 0) and the radius is \(\sqrt{7}\).

Step 1: Identify the General Form of the Circle Equation

The given equation is \( x^2 + y^2 = 7 \). This equation is in the standard form of a circle equation, which is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center of the circle and \( r \) is the radius.

Step 2: Determine the Center of the Circle

Since the given equation \( x^2 + y^2 = 7 \) can be rewritten in the form \( (x - 0)^2 + (y - 0)^2 = 7 \), we can see that the center of the circle is \((0, 0)\).

Step 3: Calculate the Radius of the Circle

From the equation \( (x - 0)^2 + (y - 0)^2 = 7 \), we see that \( r^2 = 7 \). Therefore, the radius \( r \) is equal to \( \sqrt{7} \).

Key Concepts

Center of a Circle

The center of a circle is a fundamental concept when studying the geometry of circles. It is the point that is equidistant from all points on the circumference of the circle. In the given equation, the center is derived from the general form of a circle equation, which is \((x - h)^2 + (y - k)^2 = r^2\). Here:

• \(h\) is the x-coordinate of the center
• \(k\) is the y-coordinate of the center

For the equation \(x^2 + y^2 = 7\), you can think of it as \((x - 0)^2 + (y - 0)^2 = 7\). That means the center of the circle is at \((0, 0)\). This point is right in the middle of the circle, marking the balance point around which the circle is perfectly symmetrical.

Radius of a Circle

The radius of a circle is the distance from the center of the circle to any point on its circumference. Knowing the radius helps in understanding the size and scale of the circle. It is given in the standard form equation by \(r\), where the equation is structured as \((x - h)^2 + (y - k)^2 = r^2\). The given equation, \(x^2 + y^2 = 7\), can also be rewritten as \((x - 0)^2 + (y - 0)^2 = 7\), indicating that:

• \(r^2 = 7\)

To find \(r\), you take the square root of \(7\), which results in \(r = \sqrt{7}\). The radius \(\sqrt{7}\) is an irrational number, meaning it doesn’t result in a perfect whole number or a straightforward fraction. However, it gives us the exact length from the center to the edge of the circle in this specific case.

Standard Form of a Circle

The standard form of a circle in coordinate geometry is a special type of equation that expresses all the essential features of a circle. This form is written as \((x - h)^2 + (y - k)^2 = r^2\), where:

• \((h, k)\) is the center of the circle
• \(r\) is the radius

Using the standard form makes it easy to spot both the center and the radius at a glance. The form clearly specifies the circle's position and size, offering a neat way to express circular data. For example, in the equation \(x^2 + y^2 = 7\), it relates directly to the standard form:

• \(h = 0, k = 0\)
• \(r^2 = 7\)

This way, one can immediately identify that the circle is centered at the origin and has a radius of \(\sqrt{7}\). Understanding this form simplifies manipulation and comprehension of circles in various coordinate-system related contexts.