Circle Equation: Center At Origin, Radius 6

Hey math enthusiasts! Today, we're diving into the world of circles and their equations. Specifically, we're going to figure out the canonical equation of a circle that's centered right at the origin (that's the (0,0) point on the graph) and has a radius of 6. Now, you might be thinking, "Canonical equation? That sounds complicated!" But trust me, guys, it's not as scary as it sounds. We'll break it down step by step, and by the end of this article, you'll be a pro at writing circle equations.

Understanding the Basics: What is a Circle's Canonical Equation?

Let's kick things off by understanding what a canonical equation actually is. In the realm of geometry, the canonical equation of a circle is a standard form that neatly describes a circle's properties. This form makes it super easy to identify the circle's center and radius, which are the two key ingredients you need to fully define a circle. The general form of the canonical equation for a circle centered at (h, k) with a radius of r is:

(x - h)² + (y - k)² = r²

Where:

• (x, y) represents any point on the circle's circumference.
• (h, k) represents the coordinates of the center of the circle.
• r represents the radius of the circle.

This equation might look a bit intimidating at first glance, but it's really just a clever way of expressing the Pythagorean theorem in the context of a circle. Imagine drawing a right triangle from the center of the circle to any point on its edge. The legs of the triangle would be (x - h) and (y - k), and the hypotenuse would be the radius, r. The Pythagorean theorem (a² + b² = c²) then translates directly into the canonical equation. When we talk about a circle centered at the origin, things get even simpler. Since the origin has coordinates (0, 0), we can substitute h = 0 and k = 0 into the general equation. This gives us a simplified canonical equation for circles centered at the origin:

x² + y² = r²

See? Much cleaner and easier to work with! This form tells us that if we square the x-coordinate of any point on the circle, square the y-coordinate, and add them together, we'll always get the square of the radius. This is the magic formula we'll use to find the equation of our circle.

Applying the Formula: Finding the Equation for Our Circle

Now that we've got the basics down, let's get to the heart of the problem: finding the canonical equation of our specific circle. We know that our circle is centered at the origin (0, 0) and has a radius of 6. This is all the information we need! We'll use the simplified canonical equation for circles centered at the origin:

x² + y² = r²

We know the radius, r, is 6. So, let's substitute that value into the equation:

x² + y² = 6²

Now, we just need to simplify 6²:

x² + y² = 36

And there you have it! The canonical equation of a circle centered at the origin with a radius of 6 is x² + y² = 36. This equation tells us everything we need to know about this circle. If we wanted to, we could graph this equation, and we'd see a perfect circle neatly centered at the point (0, 0) and stretching out 6 units in every direction.

Why is the Canonical Equation Useful?

You might be wondering,