Finding The Equation Of A Circle Centered At (-4, 0) Passing Through (-2, 1)

Hey everyone! Let's dive into the fascinating world of circles and their equations. Today, we're going to tackle a classic problem: finding the equation of a circle. But not just any circle – we're dealing with one that has its center precisely at the point (-4, 0) and gracefully passes through another point at (-2, 1). Sounds intriguing, right? Trust me, it's like unlocking a secret code to the circle's identity!

Understanding the Circle's Equation

Before we jump into the calculations, let's quickly refresh our understanding of what a circle's equation actually represents. You see, a circle is defined as the set of all points that are equidistant from a central point. This distance, my friends, is what we call the radius. The equation of a circle is simply a mathematical way to express this relationship. In its standard form, the equation looks like this:

(x - h)² + (y - k)² = r²

Where:

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

Think of it like a blueprint! If you know the center and the radius, you can plug those values into the equation, and you've got the circle's unique identifier. It's like having the secret formula to draw that perfect circle every single time.

Now, with this foundational knowledge in our toolkit, let's get our hands dirty and apply it to the problem at hand. Our mission is to find the equation of the circle with a center at (-4, 0) and passing through the point (-2, 1). So, let's roll up our sleeves and dive into the nitty-gritty!

Step 1: Identifying the Center (h, k)

The good news, guys, is that the first piece of the puzzle is already given to us! The problem clearly states that the center of our circle is located at the point (-4, 0). This is super helpful because it directly gives us the values for h and k in our standard equation. Remember, (h, k) represents the center, so:

• h = -4
• k = 0

See? That wasn't so bad! We've already got half of the information we need to construct the equation. It's like finding the first key to unlock a treasure chest. Now, with the center securely in our grasp, let's move on to the next crucial step: finding the radius. This is where things get a little more interesting, but don't worry, we'll break it down together and make it super clear.

Step 2: Calculating the Radius (r)

Ah, the radius! It's the lifeline of our circle, the distance that defines its size and shape. We don't have the radius explicitly given in the problem, but we do have a secret weapon: a point that lies on the circle's circumference, namely (-2, 1). This is our golden ticket to finding the radius.

Remember, the radius is the distance between the center of the circle and any point on its edge. So, to find the radius, we can simply calculate the distance between our center (-4, 0) and the point (-2, 1). And how do we calculate the distance between two points? That's right, we call upon our trusty friend, the distance formula!

The distance formula is derived from the Pythagorean theorem and looks like this:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

• d represents the distance between the two points.
• **(x₁, y₁) **and (x₂, y₂) are the coordinates of the two points.

Now, let's plug in our values:

• (x₁, y₁) = (-4, 0) (the center)
• (x₂, y₂) = (-2, 1) (the point on the circle)

Substituting these values into the distance formula, we get:

d = √[(-2 - (-4))² + (1 - 0)²]

Let's simplify this step-by-step:

d = √[(2)² + (1)²] d = √[4 + 1] d = √5

So, there you have it! The radius (r) of our circle is √5. We've conquered the distance formula and unearthed another crucial piece of our puzzle. Now, with both the center and the radius in hand, we're just one step away from revealing the full equation of the circle.

Step 3: Constructing the Equation

Alright, guys, this is the moment we've been waiting for! We've gathered all the necessary ingredients, and now it's time to bake our circular cake – or, in mathematical terms, construct the equation of our circle. Remember the standard form of the circle's equation:

(x - h)² + (y - k)² = r²

We know:

• h = -4
• k = 0
• r = √5

Let's plug these values into the equation:

(x - (-4))² + (y - 0)² = (√5)²

Now, let's simplify it:

(x + 4)² + y² = 5

And there you have it! The equation of the circle with a center at (-4, 0) and passing through the point (-2, 1) is:

(x + 4)² + y² = 5

We did it! We successfully navigated the world of circles, utilized the distance formula, and pieced together the equation that defines our specific circle. Give yourselves a pat on the back – you've earned it!

Visualizing the Circle

Now that we have the equation, let's take a moment to visualize what this circle actually looks like. Imagine a coordinate plane, like a map with two axes. Our circle's center is located at the point (-4, 0), which means it's 4 units to the left of the origin (the point where the axes cross) and right on the x-axis.

The radius, √5, is approximately 2.24 units. So, imagine drawing a circle with a center at (-4, 0) and extending 2.24 units in every direction. The point (-2, 1) would lie perfectly on the edge of this circle, confirming that our equation is indeed accurate.

Visualizing the circle helps us solidify our understanding and appreciate the connection between the equation and the geometric shape it represents. It's like seeing the blueprint come to life!

Alternative Forms of the Equation

While the standard form (x + 4)² + y² = 5 is perfectly valid and often preferred, it's worth noting that we can also express the equation in other forms. For example, we could expand the equation to get the general form:

x² + 8x + 16 + y² = 5

Simplifying further:

x² + y² + 8x + 11 = 0

The general form is less intuitive for immediately identifying the center and radius, but it's still a valid representation of the same circle. It's like having different languages to describe the same object – they might sound different, but they convey the same meaning.

Why This Matters: Applications of Circle Equations

You might be wondering,