Circle Equation: Center & Point Guide

Hey guys! Let's dive into the fascinating world of circles and equations. Today, we're going to break down how to find the equation of a circle when we know its center and a point it passes through. It might sound tricky, but trust me, it's super manageable once you get the hang of it. We'll tackle a specific problem: a circle centered at the point (5, -4) that passes through the point (-3, 2). Our mission? To fill in the blanks and nail the equation of this circle.

Understanding the Circle Equation

First things first, let's refresh our memory on the standard equation of a circle. This is our secret weapon in solving this puzzle. 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

Think of this equation as a blueprint. If we know the center (h, k) and the radius (r), we can plug those values into the equation, and boom, we've got the equation for that specific circle!

In our problem, we already know the center of the circle is (5, -4). That's a fantastic start! This means we already have our 'h' and 'k' values. The next piece of the puzzle is the radius, 'r'. Remember, the radius is the distance from the center of the circle to any point on the circle. We're given that the circle passes through the point (-3, 2), which means we can use this point to figure out the radius.

So, the first key step in finding the circle's equation is identifying what the standard form represents: the relationship between a circle's center coordinates (h, k), its radius (r), and any point (x, y) lying on the circle. Grasping this foundational formula, (x - h)² + (y - k)² = r², is crucial. It's not just a jumble of letters and symbols; it's a powerful tool that geometrically expresses the Pythagorean theorem applied to the circle. Imagine drawing a right-angled triangle where the hypotenuse is the radius, and the other two sides are the horizontal and vertical distances from the center to a point on the circumference. This visualization helps to embed the equation in your understanding.

The next challenge is to extract the information given in the problem. We're told that our circle has its center at the point (5, -4). This immediately gives us the values for h and k, which are 5 and -4, respectively. Writing these down clearly is a good practice: h = 5, k = -4. Misidentifying these values, even by a sign error, will lead to an incorrect equation. So, double-check that you've correctly matched the coordinates to their respective variables in the standard equation. Now, we're halfway there in terms of plugging into the left-hand side of the equation. We've got the (x - h)² and (y - k)² parts almost ready. We just need to substitute our h and k values. But before we do, let's focus on the other piece of information we have: the circle passes through the point (-3, 2). This is our key to unlocking the radius, r, which is the last piece of the puzzle needed to fully define the circle's equation.

Calculating the Radius

Now, how do we find the radius? Here's where another important concept comes into play: the distance formula. The distance formula helps us calculate the distance between two points in a coordinate plane. Since the radius is the distance between the center of the circle (5, -4) and the point on the circle (-3, 2), we can use the distance formula:

r = √[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

• (x₁, y₁) is the center of the circle (5, -4)
• (x₂, y₂) is the point on the circle (-3, 2)

Let's plug in the values:

r = √[(-3 - 5)² + (2 - (-4))²] r = √[(-8)² + (6)²] r = √(64 + 36) r = √100 r = 10

So, the radius of our circle is 10! We're on a roll!

The radius, r, is the distance from the center of the circle to any point on its circumference. We have the center (5, -4) and a point on the circle (-3, 2). The most direct route to finding r is by employing the distance formula, which is derived from the Pythagorean theorem. Think of it as constructing a right-angled triangle with the radius as the hypotenuse and the differences in the x and y coordinates as the sides. The distance formula, √[(x₂ - x₁)² + (y₂ - y₁)²], formalizes this relationship.

Substituting our points (5, -4) and (-3, 2) into the formula is the next step. Let's break it down: x₂ is -3, x₁ is 5, y₂ is 2, and y₁ is -4. Plugging these in, we get √[(-3 - 5)² + (2 - (-4))²]. It's crucial to handle the subtraction of negative numbers carefully here. A common mistake is to miscalculate 2 - (-4) as 2 - 4 = -2, rather than the correct 2 + 4 = 6. Attention to detail in these arithmetic steps is essential for accuracy. Once we've correctly substituted, we simplify the expressions inside the parentheses: (-3 - 5) becomes -8, and (2 - (-4)) becomes 6. Now we have √[(-8)² + (6)²].

Next, we square -8 and 6, resulting in 64 and 36, respectively. Remember that squaring a negative number always yields a positive result. This is a frequent area for errors if negative signs are mishandled. So, we now have √(64 + 36). Adding these gives us √100. The final step is to take the square root of 100, which is 10. Therefore, the radius, r, of our circle is 10. This value represents the constant distance from the center of the circle to any point on its edge. Now that we've calculated the radius, we have all the pieces needed to assemble the complete equation of the circle. We have the center (h, k) = (5, -4) and the radius r = 10. The only remaining step is to plug these values into the standard equation and simplify.

Plugging in the Values

We have all the information we need! Let's plug the values of h, k, and r into the standard equation of a circle:

(x - h)² + (y - k)² = r²

(x - 5)² + (y - (-4))² = 10²

(x - 5)² + (y + 4)² = 100

Notice how we handled the negative sign in front of the 'k' value. Subtracting a negative is the same as adding a positive, so (y - (-4)) becomes (y + 4).

Substituting the values we've found into the standard form is the culmination of our efforts. We have h = 5, k = -4, and r = 10. The standard equation is (x - h)² + (y - k)² = r². Carefully placing these values into their respective positions gives us (x - 5)² + (y - (-4))² = 10². The key here is to substitute correctly, paying close attention to signs. A common mistake is to mix up the values or to neglect the negative sign when substituting k = -4. So, double-checking this step is always a good idea. The next substep involves simplifying the equation. We have (x - 5)² + (y - (-4))² = 10². We can simplify (y - (-4)) to (y + 4) since subtracting a negative number is the same as adding. Also, 10² is simply 10 * 10, which equals 100. Making these simplifications, our equation becomes (x - 5)² + (y + 4)² = 100. This is the equation of the circle in standard form. It directly tells us the center and the radius of the circle.

Now, let’s address the original question format, which asked us to fill in the blanks. The equation we've arrived at, (x - 5)² + (y + 4)² = 100, needs to be slightly adjusted to match the format (x + □)² + (y + □)² = □. We already have the (y + 4)² part matching perfectly. For the (x - 5)² term, we need to think about what number, when added, would give us the same result as subtracting 5. This is where we recognize that adding the negative of a number is the same as subtracting the number. So, (x - 5) is the same as (x + (-5)). This gives us the form (x + □)², where the missing number is -5. For the right-hand side of the equation, we already have the simplified form 100, which is the square of the radius. So, no further adjustment is needed there. By carefully considering the format of the question and making these slight adjustments, we can confidently fill in the blanks with the correct values. This ensures we provide the answer in the exact form requested, demonstrating a thorough understanding of the problem and its solution.

Filling in the Blanks

Now, let's go back to the original question format. We need to fill in the boxes in this equation:

(x + $\square$ )² + (y + $\square$)² = $\square$

Comparing this to our equation, (x - 5)² + (y + 4)² = 100, we can see:

• In the first box, we need to put the value that, when added to x, gives us the same result as (x - 5). That value is -5.
• In the second box, we already have (y + 4), so the value is 4.
• In the last box, we have the radius squared, which is 100.

So, the filled-in equation is:

(x + (-5))² + (y + 4)² = 100

Or, more simply:

(x - 5)² + (y + 4)² = 100

And there you have it! We've successfully found the equation of the circle and filled in the blanks.

The final step in our journey is to translate our mathematical understanding into the specific answer format requested. The original problem presented us with an equation skeleton: (x + □)² + (y + □)² = □, and our task is to fill in the blanks with the correct values. We've already derived the equation of the circle in standard form: (x - 5)² + (y + 4)² = 100. The challenge now is to map our results onto the given structure accurately. The first term in the skeleton is (x + □)². Comparing this with our (x - 5)², we need to recognize that subtracting 5 is equivalent to adding -5. This is a crucial point: the blank should be filled with -5, not 5. This often trips up students who might overlook the significance of the '+' sign in the skeleton. Similarly, for the second term, we have (y + □)². Our equation already has (y + 4)², so the blank here is simply 4. There's no sign manipulation needed in this case. Finally, for the right-hand side, we have □. Our equation has 100, which represents r², the square of the radius. So, the last blank is filled with 100. Therefore, the completed equation is (x + (-5))² + (y + 4)² = 100. This meticulously crafted answer not only solves the problem but also demonstrates a clear grasp of the underlying mathematical principles and attention to detail in translating the solution into the required format. This step-by-step approach ensures that no marks are lost due to careless errors in presentation.

Key Takeaways

Let's recap what we've learned:

1. The standard equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
2. We can use the distance formula to find the radius if we know the center and a point on the circle.
3. Plugging the values of h, k, and r into the standard equation gives us the equation of the circle.
4. Always pay attention to signs, especially when dealing with negative numbers!

This problem might seem daunting at first, but by breaking it down into smaller steps and understanding the underlying concepts, you can conquer it with confidence. Keep practicing, and you'll be a circle equation pro in no time!