Circle Equation Diameter 12 Center On X-Axis

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Hey guys! Let's dive into a fun geometry problem where we're figuring out the equation of a circle. We know the circle has a diameter of 12 units, and the really important part is that its center sits right on the x-axis. This little tidbit gives us a huge clue about what the equation should look like. We've got a bunch of options, and our mission is to pick the two that fit the bill perfectly.

Understanding the Circle Equation

Before we jump into the options, let's quickly recap the standard form equation of a circle. This is our secret weapon for solving this problem! The equation looks like this:

(x - h)² + (y - k)² = r²

Where:

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

Knowing this, we can dissect the information given in the problem and see how it translates into this equation. The problem states that the circle has a diameter of 12 units. Remember, the radius is half the diameter, so our radius (r) is 6 units. This means r² will be 6² = 36. So, one part of our equation is already solved! The equation will end with "= 36".

Now, the center of the circle is the next key piece. We know the center lies on the x-axis. What does that tell us? It means the y-coordinate of the center (k) is 0. Any point on the x-axis has a y-coordinate of 0. The x-coordinate (h) can be any number, as long as it places the circle somewhere along the x-axis. This is where our options come in handy, as they will show us different possible x-coordinates for the center.

So, to recap, we're looking for equations that have the form: (x - h)² + y² = 36. The value of 'h' will determine where the center lies on the x-axis, and that's what we need to figure out from the choices we're given. Remember, the sign in the equation is opposite to the sign of the coordinate. For example, if we have (x - 6)², the x-coordinate of the center is actually +6. Similarly, (x + 6)² means the x-coordinate is -6. Keep this in mind as we analyze each option. This is a common trick in these types of problems, so mastering this will help you a lot in the future!

Analyzing the Options

Okay, let's put on our detective hats and examine each equation to see if it fits our criteria. We're looking for two equations, so we'll go through each one systematically. Remember, our target equation form is (x - h)² + y² = 36.

Option 1: (x - 12)² + y² = 1

Let's break this down. The center of this circle would be at (12, 0), which is on the x-axis – that part is correct! However, the radius squared (r²) is 1, meaning the radius is 1. But we know our radius needs to be 6 because the diameter is 12. So, this option is a no-go. It fails the radius test.

Option 2: x² + y² = 12

This one is interesting. We can rewrite it as (x - 0)² + (y - 0)² = 12. This tells us the center is at (0, 0), which is indeed on the x-axis. But again, the radius squared is 12, meaning the radius is √12 (approximately 3.46), not 6. So, this option is also incorrect. It's close on the center but fails on the radius.

Option 3: (x + 6)² + y² = 36

Now we're talking! The center here is (-6, 0), which is perfectly situated on the x-axis. And, crucially, the radius squared is 36, meaning the radius is 6 – exactly what we need! This equation fits all our criteria. So, this is one of our correct answers. Awesome!

Option 4: (x - 6)² + y² = 36

This one looks promising too! The center is (6, 0), which is on the x-axis. And just like option 3, the radius squared is 36, giving us a radius of 6. This is another winner! We've found our second correct answer.

Option 5: (x + 12)² + y² = 1

This option has a center at (-12, 0), which is on the x-axis. But, just like option 1, the radius squared is 1, making the radius 1. This doesn't match our required radius of 6, so it's incorrect.

Option 6: x² + y² = 144

This equation has a center at (0, 0), which lies on the x-axis. However, the radius squared is 144, meaning the radius is 12. This is double the radius we need, so this option is incorrect.

Conclusion: The Correct Equations

Alright, we've done the detective work, and we've cracked the case! The two equations that represent a circle with a diameter of 12 units and a center on the x-axis are:

  • (x + 6)² + y² = 36
  • (x - 6)² + y² = 36

These equations both have a radius of 6 (since 6² = 36) and centers at (-6, 0) and (6, 0), respectively, both of which lie on the x-axis. So, there you have it! We successfully navigated the world of circle equations and found the solutions. Remember, understanding the standard form of the equation and how the center and radius relate to it is key to tackling these kinds of problems. Keep practicing, and you'll become a circle equation master in no time!

Understanding Circle Equations A Comprehensive Guide

Alright, let's talk circles! Specifically, we're going to dissect the equation of a circle, that fundamental shape that pops up everywhere from pizzas to planets. This might seem like a dry math topic, but trust me, understanding circle equations is like unlocking a secret code to a whole world of geometry. It’s not just about memorizing formulas; it’s about grasping the underlying concepts. So, grab your metaphorical compass and straightedge, and let’s dive in! The goal here is to not only understand but to really feel how these equations work.

The Anatomy of a Circle: Center and Radius

Before we even peek at an equation, let's nail down the basics. What defines a circle? Two key elements: the center and the radius. The center is that special point smack-dab in the middle of the circle. It's like the circle's home base. The radius is the distance from the center to any point on the circle's edge. Think of it as the circle's reach. Now, imagine you're drawing a circle. You plant your compass at the center, and the radius is the length you set on the compass. As you spin the compass, it traces out all the points that are exactly that radius-length away from the center. That’s what a circle is all about: a collection of points equidistant from a central point.

The Standard Form Equation: Unveiling the Code

Okay, now for the main event: the equation! The standard form equation of a circle is your golden ticket to understanding and manipulating circles in the coordinate plane. It looks like this:

(x - h)² + (y - k)² = r²

Whoa, hold up! Don't let those letters scare you. Each one has a specific role to play, and once you understand them, the equation becomes your friend. Let's break it down:

  • (x, y): These are the coordinates of any point that lies on the circle. It's like saying,