Finding The Equation Of A Circle Passing Through A Point
Hey guys! Let's dive into a fun math problem today where we'll figure out how to find the equation of a circle. We're given a point that the circle passes through and the center of the circle. Sounds like a geometric adventure, right? Let's get started!
The Basics of Circle Equations
Before we jump into the problem, let's quickly refresh our memory on the standard form of a circle's equation. This is super important because it's the key to solving our 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 already see how the center of the circle plays a role in the equation. But what about the radius? That's where our given point comes in!
The Problem: A Circle Through (-2, 8) with Center (4, 0)
Here's the question we need to crack: Which equation represents a circle that contains the point (-2, 8) and has a center at (4, 0)?
We've got a few options to choose from:
- (x - 4)² + y² = 100
- (x - 4)² + y² = 10
- x² + (y - 4)² = 10
- x² + (y - 4)² = 100
To solve this, we need to figure out which equation fits both the center and the point given.
Step 1: Plugging in the Center
Our circle's center is at (4, 0). So, in our standard equation (x - h)² + (y - k)² = r², we know that h = 4 and k = 0. Let's plug these values in:
(x - 4)² + (y - 0)² = r²
Simplifying this a bit, we get:
(x - 4)² + y² = r²
Now, looking at our options, we can eliminate the ones that don't have this form. Specifically, we can rule out the equations x² + (y - 4)² = 10 and x² + (y - 4)² = 100 because they have the wrong form for the center. This leaves us with:
- (x - 4)² + y² = 100
- (x - 4)² + y² = 10
Step 2: Finding the Radius
We're halfway there! Now we need to figure out which of the remaining equations has the correct radius. This is where the point (-2, 8) comes into play. Remember, this point lies on the circle, so it must satisfy the circle's equation.
To find the radius, we'll use the distance formula. The distance formula helps us find the distance between two points, which in our case is the distance between the center of the circle (4, 0) and the point on the circle (-2, 8). This distance is the radius.
The distance formula is:
√[(x₂ - x₁)² + (y₂ - y₁)²]
Let's plug in our points:
- (x₁, y₁) = (4, 0) (the center)
- (x₂, y₂) = (-2, 8) (the point on the circle)
So, the radius (r) is:
r = √[(-2 - 4)² + (8 - 0)²] r = √[(-6)² + (8)²] r = √[36 + 64] r = √100 r = 10
Great! We found that the radius of the circle is 10.
Step 3: Squaring the Radius and Matching the Equation
Remember, in the circle's equation, we use r², not just r. So, we need to square our radius:
r² = 10² r² = 100
Now we know that the right side of our circle's equation should be 100. Looking back at our options, the correct equation is:
(x - 4)² + y² = 100
Therefore, the answer is (x - 4)² + y² = 100.
We've successfully found the equation of the circle! We used the center to narrow down our options and then used the distance formula and the given point to calculate the radius. By squaring the radius, we matched it to the correct equation.
Alternative Method: Plugging the Point into the Equations
There's another way to solve this problem! Instead of finding the radius directly, we could have plugged the point (-2, 8) into the remaining equations after using the center to narrow down the choices.
Let's try it with the two equations we had left:
- (x - 4)² + y² = 100
- (x - 4)² + y² = 10
Plugging in x = -2 and y = 8 into the first equation:
(-2 - 4)² + (8)² = 100 (-6)² + 64 = 100 36 + 64 = 100 100 = 100
This equation holds true! Now let's try plugging the point into the second equation:
(-2 - 4)² + (8)² = 10 (-6)² + 64 = 10 36 + 64 = 10 100 = 10
This equation is not true. So, we confirm that the correct equation is (x - 4)² + y² = 100.
Key Takeaways
- The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
- The distance formula helps find the distance between two points, which can be used to find the radius of a circle if you know the center and a point on the circle.
- A point on the circle must satisfy the circle's equation. You can plug the point's coordinates into the equation to check if it's correct.
Practice Makes Perfect
Finding the equation of a circle is a fundamental concept in geometry. By understanding the standard form of the equation and using tools like the distance formula, you can solve these problems with confidence. So, keep practicing, and you'll become a circle-equation pro in no time!
Conclusion
We've walked through the steps to find the equation of a circle given its center and a point on the circle. Whether you prefer using the distance formula to find the radius or plugging the point directly into the equations, you now have the tools to tackle similar problems. Keep exploring the world of geometry, and you'll discover even more exciting mathematical concepts!