Circle Equation How To Find Equation Of Circle

Hey guys! Let's dive into a cool math problem involving circles. We're going to figure out how to find the equation of a circle when we know its center and a point it passes through. Trust me, it's not as scary as it sounds! We'll break it down step-by-step, so you'll be a circle equation whiz in no time. Think of it as unlocking a secret code to the world of geometry! So, grab your pencils, and let's get started on this mathematical adventure!

The Circle Challenge: Finding the Equation

Okay, so here's the problem we're tackling: Imagine a circle, we'll call it circle C, hanging out on a coordinate plane. This circle has its center at a specific spot, the point (-2, 10). Now, this circle isn't just floating in space; it's also passing through another point, P, located at (10, 5). Our mission, should we choose to accept it (and we do!), is to find the equation that perfectly describes circle C. We've got a couple of options to choose from, and only one is the real deal:

A. $(x-2)^2+(y+10)2=13$ B. $(x-2)^2+(y+10)2=169$ C. $(x+2)^2+(y-10)2=169$ D. $(x+2)^2+(y-10)2=13$

Which one do you think it is? Don't worry if you're not sure yet. We're going to walk through the process together. The key here is understanding what the equation of a circle actually tells us. It's like a treasure map, where the equation gives us the coordinates to find the circle's center and its size (the radius). We'll use the information we have – the center and the point on the circle – to decode the equation. Think of it as a mathematical puzzle, and we're the detectives cracking the case!

Unpacking the Circle Equation

Before we jump into solving our specific problem, let's make sure we're all on the same page about what the equation of a circle actually means. This is the foundation, guys, the secret sauce that makes everything else click. The standard form equation of a circle is written as:

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

Now, let's break this down piece by piece. It might look like a jumble of letters and symbols, but each one has a specific job:

• (x, y): These are the variables that represent any point on the circle. Think of them as the coordinates that trace the circle's path as you go around it. They're the dynamic duo, always changing as you move along the circle's edge.
• (h, k): This is the superstar of the equation – the center of the circle! The coordinates (h, k) tell you exactly where the circle is located on the coordinate plane. It's the anchor point, the heart of the circle.
• r: This little guy represents the radius of the circle. The radius is the distance from the center of the circle to any point on its edge. It's what determines how big or small the circle is.
• r^2: Notice that in the equation, we have r squared. This is important because we'll often need to find the radius first and then square it to plug it into the equation.

So, the equation is basically a recipe. If you know the center (h, k) and the radius r, you can plug those values into the equation, and you've got the unique fingerprint of that circle. Conversely, if you're given the equation, you can immediately read off the center and the radius. It's like being able to read the circle's mind!

Cracking the Code: Finding the Radius

Alright, now that we're fluent in circle equation language, let's get back to our challenge. We know the center of circle C is at (-2, 10), and it passes through the point P (10, 5). We've got the (h, k) part of the equation puzzle solved – that's the center. What we're missing is r, the radius. But don't worry, we have a secret weapon: the distance formula!

The distance formula is our trusty tool for finding the distance between two points on a coordinate plane. Remember, the radius is simply the distance between the center of the circle and any point on the circle. So, if we can find the distance between (-2, 10) and (10, 5), we've found our radius.

The distance formula looks like this:

$$Distance = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Where:

• (x₁, y₁) are the coordinates of the first point (in our case, the center of the circle).
• (x₂, y₂) are the coordinates of the second point (the point P on the circle).

Let's plug in our values:

• (x₁, y₁) = (-2, 10)
• (x₂, y₂) = (10, 5)

So, the distance (which is our radius, r) becomes:

$$r = \sqrt{(10 - (-2))^2 + (5 - 10)^2}$$

Now, let's simplify this step-by-step. First, handle the stuff inside the parentheses:

$$r = \sqrt{(10 + 2)^2 + (5 - 10)^2}$$

$$r = \sqrt{(12)^2 + (-5)^2}$$

Next, square those numbers:

$$r = \sqrt{144 + 25}$$

And finally, add them up and take the square root:

$$r = \sqrt{169}$$

$$r = 13$$

Boom! We've found the radius. It's 13 units. We're one giant step closer to cracking the circle equation code!

Completing the Puzzle: Plugging in the Values

Okay, guys, we've done the hard work. We know the center of the circle (h, k) is (-2, 10), and we've calculated the radius r to be 13. Now comes the satisfying part: plugging these values into the standard form equation of a circle:

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

Let's substitute our values:

• h = -2
• k = 10
• r = 13

Plugging these in, we get:

$$(x - (-2))^2 + (y - 10)^2 = 13^2$$

Now, let's simplify this a bit. Subtracting a negative is the same as adding, and 13 squared is 169:

$$(x + 2)^2 + (y - 10)^2 = 169$$

And there you have it! This is the equation that perfectly describes circle C. It's the unique fingerprint of this particular circle, telling us exactly where it is and how big it is. We've successfully decoded the circle's secrets!

The Grand Reveal: Choosing the Correct Answer

Now that we've found the equation of circle C, let's go back to our original options and see which one matches. We were given these choices:

A. $(x-2)^2+(y+10)2=13$ B. $(x-2)^2+(y+10)2=169$ C. $(x+2)^2+(y-10)2=169$ D. $(x+2)^2+(y-10)2=13$

We found the equation to be:

$$(x + 2)^2 + (y - 10)^2 = 169$$

Looking at the options, we can clearly see that option C is the winner! It's the exact equation we derived, matching the center and radius of circle C perfectly. Option C is the correct answer.

Why the Others Don't Fit

It's always a good idea to understand not just why the correct answer is right, but also why the incorrect answers are wrong. This helps solidify our understanding of the concepts.

• Option A: $(x-2)^2+(y+10)2=13$ This option has the wrong signs for the center coordinates (it should be +2 and -10, not -2 and +10) and the wrong radius squared (it should be 169, not 13).
• Option B: $(x-2)^2+(y+10)2=169$ This option has the correct radius squared (169), but the signs for the center coordinates are still incorrect.
• Option D: $(x+2)^2+(y-10)2=13$ This option has the correct signs for the center coordinates, but the radius squared is incorrect (it should be 169, not 13).

By analyzing why these options are wrong, we reinforce our understanding of how each part of the circle equation contributes to its overall representation. It's like seeing the puzzle from all angles, making us even better at solving circle equation challenges in the future.

Wrapping Up: Circle Equation Superpowers Acquired!

Great job, everyone! You've successfully navigated the world of circle equations. We took a problem, broke it down step-by-step, and emerged victorious. You now have the power to find the equation of a circle given its center and a point on the circle. That's a pretty awesome superpower to have!

Remember, the key is understanding the standard form equation of a circle and how each component (the center and the radius) plays its part. With a little practice, you'll be spotting circle equations in your sleep (okay, maybe not in your sleep, but you'll definitely feel confident tackling them!).

So, the next time you see a circle equation problem, don't sweat it. Just remember the steps we've covered: identify the center, find the radius (using the distance formula if needed), and plug those values into the equation. You've got this! Keep practicing, keep exploring, and keep those mathematical muscles strong!