Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the circle with the given characteristics. Center at origin; radius: \(\sqrt{11}\)

Step-by-Step Solution

Answer

The standard form of the equation of the circle with given characteristics is \(x^2 + y^2 = 11\).

Step 1: Identify the given parameters

The center of the circle is at the origin, and the radius of the circle is \(\sqrt{11}\). So, we have the center coordinates \((0,0)\) and radius \(r = \sqrt{11}\).

Step 2: Apply the values in the circle's equation

The equation of a circle with center at the origin in two-dimensional space is given by \(x^2 + y^2 = r^2\). Substituting our given radius \(\sqrt{11}\) into the equation, we get \(x^2 + y^2 = (\sqrt{11})^2\).

Step 3: Simplify the equation

On simplifying, \((\sqrt{11})^2\) gives us 11. Therefore, the equation of the circle in standard form is \(x^2 + y^2 = 11\).

Key Concepts

Center at Origin

Whenever you learn about equations of circles, the location of the circle's center is crucial. When we talk about a circle with its **center at the origin**, we're referring to a specific point on a coordinate plane. The origin is the point

• where the x-axis and y-axis intersect,
• and it is represented by the coordinates \((0, 0)\).

This location makes writing the equation of the circle simpler compared to circles centered elsewhere.
Any other points on the circle are at a consistent distance, known as the radius, from this center point.
The simplicity of the origin as a center means the equation doesn't need to account for shifts along the x or y-axis. This is why the circle's formula becomes more straightforward, without additional terms to compensate for location shifts.

Radius

Understanding the **radius** is key to diving deeper into the mathematics of a circle. The radius is the distance from the center of the circle to any point on its boundary.
In this exercise, the radius is given as \(\sqrt{11}\). This tells us that:

• Every point on the circle's outline is exactly \(\sqrt{11}\) units away from the origin.

Because the radius affects the size of the circle, knowing how to use it in the circle's equation is important.
The squared radius,\((\sqrt{11})^2\), uses simple arithmetic to turn the radial distance into \(11\).
In essence, it acts as a scalar factor in the equation, shaping the size of the circle on the coordinate plane.

Standard Form of a Circle

One of the most common ways to represent a circle in coordinates is by using its **standard form** equation.
This standard form helps to easily identify the center and radius of the circle. For a circle centered at the origin, the equation simplifies to:

• \[x^2 + y^2 = r^2\],

where \(r\) is the radius.
For this specific exercise, since the center is at \((0, 0)\) and \(r = \sqrt{11}\),
we substitute \(r^2\) with \(11\):

• The equation becomes\[x^2 + y^2 = 11.\]

This form is both simple and powerful, providing everything needed to graph the circle and understand its geometry instantly.