Problem 44 - Algebra A Combined Function

Question

Write an equation of the circle with the given center and radius. See Example 8. The origin; \(4 \sqrt{7}\)

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 = 112\).

Step 1: Identify the components

Identify the center and radius of the circle. Here, the center is at the origin \((0, 0)\) and the radius is \(4 \sqrt{7}\).

Step 2: Use the general circle equation

The general equation of a circle with center \((h, k)\) and radius \(r\) is given by: \((x - h)^2 + (y - k)^2 = r^2\).

Step 3: Substitute values into the equation

Substitute the center \((0, 0)\) and radius \(4 \sqrt{7}\) into the general equation: \((x - 0)^2 + (y - 0)^2 = (4 \sqrt{7})^2\).

Step 4: Calculate the radius squared

Calculate \((4 \sqrt{7})^2\), which equals \(16 \cdot 7 = 112\).

Step 5: Write the final equation of the circle

Using the values from previous steps, the equation of the circle is \(x^2 + y^2 = 112\).

Key Concepts

Center of a Circle

In geometry, the center of a circle is a fundamental concept.
It is the fixed point from which every point on the circle is equidistant.
This distance is known as the radius.
For any circle, when you perfectly draw it, all points around the edge are exactly the same distance from the center.
When we talk about the circle equation, knowing the center is critical.
The generic equation of a circle has the form \[(x - h)^2 + (y - k)^2 = r^2,\] where

• \((h, k)\) represents the coordinates of the center.
• \(r\) is the radius.

This notation means that you can shift the circle around on a grid by changing the \((h, k)\) values. The example discussed uses the origin \((0, 0)\) as the center.
At the origin, the center is at the intersection of the x-axis and y-axis.
This simplifies the circle equation because we don't need to adjust for an offset. The equation reflects the property of the circle centered at the origin: \[x^2 + y^2 = r^2.\]

Radius of a Circle

The radius of a circle is quite simply the distance from the center of the circle to any point along its perimeter.
The length of the radius is constant throughout the circle.Having a fixed center point, you can draw a line (the radius) to form the circle.In the general circle equation \((x - h)^2 + (y - k)^2 = r^2,\) \(r\) represents the radius.
This value is squared in the equation to show its relationship with the points on the perimeter.In our exercise, the radius is given as \(4 \sqrt{7}\).
To use this in the formula, simply square it:

• Calculate \((4 \sqrt{7})^2\)
• which results in \(16 \cdot 7 = 112\).

Thus, for a circle with this radius, every point on the circle's edge is 4\( \sqrt{7} \) units from the center.
This value, squared, gives the term \(r^2\) which we plug into our circle equation:

Geometry

Geometry is the branch of mathematics concerning shapes, sizes, and the properties of space.
Circles are one of the most fundamental shapes in geometry. Understanding the equation of a circle is key to understanding more complex geometric properties.
The primary components of a circle include:

• The center, which is the anchor for constructing a circle.
• The radius, which determines the size of the circle.
• The circumference, which is the distance around the circle.
• The area, which is the space contained within the circle.

Circles play a crucial role in geometry, often serving as a basis for exploring other shapes and mathematical concepts.
By analyzing how a circle functions from its center and radius, students gain valuable insights into more advanced geometrical problems.
As you study further, you'll find many formulas and theorems relate back to a circle's basic properties.
Whether you're working on advanced algebra, calculus, or trigonometry, the geometry of a circle often appears.
Understanding its equation provides a strong foundation for success in these subjects.