Finding The Radius Of A Circle $x^2 + Y^2 + 6x - 2y + 3 = 0$

Hey there, math enthusiasts! Today, we're diving into the fascinating world of circles and their equations. Specifically, we're going to tackle the equation $x^2 + y^2 + 6x - 2y + 3 = 0$ and figure out the radius of the circle it represents. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the General Equation of a Circle

Before we jump into the nitty-gritty, let's quickly review the general equation of a circle. This will give us a solid foundation for understanding how to extract information from the given equation. The general form of a circle's equation is:

$$(x - h)^2 + (y - k)^2 = r^2$$

Where:

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

Think of this equation as a blueprint for a circle. The values of h, k, and r completely define the circle's position and size on the coordinate plane. The center coordinates (h, k) tell us where the circle is located, and the radius r tells us how big it is. A larger radius means a bigger circle, and a smaller radius means a smaller circle. It's all quite intuitive, isn't it?

Now, our mission is to transform the given equation, $x^2 + y^2 + 6x - 2y + 3 = 0$, into this familiar general form. This will allow us to easily identify the center and, most importantly, the radius of the circle. We'll achieve this transformation by using a clever technique called "completing the square." This method might sound intimidating, but trust me, it's a powerful tool that makes solving these types of problems much easier. So, let's roll up our sleeves and get started!

The Key Technique: Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in a more convenient form. In the context of circle equations, it helps us transform the given equation into the standard form, $(x - h)^2 + (y - k)^2 = r^2$.

Here's the general idea behind completing the square. Consider a quadratic expression of the form $x^2 + bx$. To complete the square, we need to add a constant term that will make the expression a perfect square trinomial. A perfect square trinomial can be factored into the form $(x + a)^2$ or $(x - a)^2$.

The constant term we need to add is $(b/2)^2$. This is because:

$$x^2 + bx + (b/2)^2 = (x + b/2)^2$$

Now, let's apply this technique to our equation, $x^2 + y^2 + 6x - 2y + 3 = 0$. The goal is to rewrite the equation by completing the square for both the x terms and the y terms. This will allow us to express the equation in the standard form of a circle's equation.

First, we'll group the x terms and the y terms together:

$$(x^2 + 6x) + (y^2 - 2y) + 3 = 0$$

Next, we'll complete the square for the x terms. The coefficient of our x term is 6, so we need to add and subtract $(6/2)^2 = 3^2 = 9$:

$$(x^2 + 6x + 9 - 9) + (y^2 - 2y) + 3 = 0$$

Now, the expression inside the first parenthesis, $x^2 + 6x + 9$, is a perfect square trinomial. We can factor it as $(x + 3)^2$:

$$((x + 3)^2 - 9) + (y^2 - 2y) + 3 = 0$$

We'll repeat the process for the y terms. The coefficient of our y term is -2, so we need to add and subtract $(-2/2)^2 = (-1)^2 = 1$:

$$((x + 3)^2 - 9) + (y^2 - 2y + 1 - 1) + 3 = 0$$

The expression inside the second parenthesis, $y^2 - 2y + 1$, is also a perfect square trinomial. We can factor it as $(y - 1)^2$:

$$((x + 3)^2 - 9) + ((y - 1)^2 - 1) + 3 = 0$$

Now, we'll rearrange the equation to group the squared terms and the constant terms:

$$(x + 3)^2 + (y - 1)^2 - 9 - 1 + 3 = 0$$

Finally, we'll simplify the constant terms:

$$(x + 3)^2 + (y - 1)^2 - 7 = 0$$

To get the equation in the standard form, we'll move the constant term to the right side of the equation:

$$(x + 3)^2 + (y - 1)^2 = 7$$

And there we have it! We've successfully transformed the original equation into the standard form of a circle's equation. Now, we can easily identify the center and the radius of the circle.

Identifying the Center and Radius

Now that we've massaged our equation into the familiar form of $(x - h)^2 + (y - k)^2 = r^2$, it's time to put on our detective hats and extract the key information: the center and the radius. Let's revisit our transformed equation:

$$(x + 3)^2 + (y - 1)^2 = 7$$

Remember, the center of the circle is represented by the coordinates (h, k), and the radius is represented by r. By comparing our transformed equation to the standard form, we can easily identify these values.

Notice that we have $(x + 3)^2$, which can be rewritten as $(x - (-3))^2$. This tells us that the x-coordinate of the center, h, is -3. Similarly, we have $(y - 1)^2$, which directly tells us that the y-coordinate of the center, k, is 1. So, the center of our circle is (-3, 1). That wasn't so hard, was it?

Now, let's focus on the right side of the equation, which is equal to $r^2$. In our case, we have $r^2 = 7$. To find the radius r, we simply need to take the square root of both sides of the equation:

$$r = \sqrt{7}$$

So, the radius of our circle is $\sqrt{7}$. However, the problem asks us to round our answer to the nearest thousandth. Let's grab our calculators and get to work!

Calculating and Rounding the Radius

We've determined that the radius of our circle is $\sqrt{7}$. Now, we need to find the approximate value of $\sqrt{7}$ and round it to the nearest thousandth. A calculator is our best friend here. Plugging in $\sqrt{7}$ into a calculator, we get approximately 2.645751311...

To round this number to the nearest thousandth, we need to look at the digit in the ten-thousandths place, which is the fourth digit after the decimal point. In this case, it's a 7. Since 7 is greater than or equal to 5, we round up the digit in the thousandths place.

The digit in the thousandths place is 5. Rounding it up gives us 6. So, rounding 2.645751311... to the nearest thousandth gives us 2.646. Remember guys, rounding is an essential skill in mathematics and everyday life, helping us to express numbers in a more manageable and understandable way.

Conclusion: The Radius Revealed

Alright, math detectives, we've cracked the case! By skillfully employing the technique of completing the square, we successfully transformed the equation $x^2 + y^2 + 6x - 2y + 3 = 0$ into the standard form of a circle's equation. This allowed us to easily identify the center of the circle as (-3, 1) and the radius as $\sqrt{7}$. Finally, we rounded the radius to the nearest thousandth, arriving at our answer of 2.646. Remember, practice makes perfect, so keep honing your skills, and you'll become a master of circle equations in no time!