Circle Equation & Graph Through 3 Points: A Math Guide
Hey everyone! Today, we're going to tackle a classic geometry problem: finding the equation and graph of a circle that passes through three given points. Specifically, we'll use the points E(-1, -8), F(5, -2), and G(11, -8). Buckle up, because we're about to dive deep into circles, equations, and a bit of algebraic manipulation. Let's get started!
Understanding the Basics
Before we jump into the solution, let's quickly recap some fundamental concepts about circles and their equations. The standard form of a circle's equation is given by:
(x - h)² + (y - k)² = r²
Where:
- (h, k) is the center of the circle.
- r is the radius of the circle.
Our main goal is to find the values of h, k, and r using the given points E, F, and G. Each point lies on the circle, which means that they must satisfy the circle's equation. This will give us three equations, which we can solve simultaneously to find our unknowns.
The approach involves substituting the coordinates of points E, F, and G into the standard equation of a circle and then solving the resulting system of equations. This might sound a bit complicated, but don't worry, we'll break it down step by step.
Now, let’s understand why three points uniquely define a circle. Imagine you have two points; you can draw infinitely many circles passing through them. However, adding a third non-collinear point restricts the circle to a unique one. This is because the center of the circle must be equidistant from all three points, which leads to a specific solution.
Remember, the key to solving this problem is to carefully substitute and simplify the equations. Accuracy in algebraic manipulation is crucial. This process will not only help you find the equation of the circle but also reinforce your understanding of coordinate geometry. So, keep your pencils sharp and your minds sharper as we embark on this mathematical journey!
Setting Up the Equations
Okay, let's start by plugging the coordinates of our points E(-1, -8), F(5, -2), and G(11, -8) into the standard circle equation:
-
For point E(-1, -8): ((-1) - h)² + ((-8) - k)² = r² (1 + h)² + (8 + k)² = r²
-
For point F(5, -2): (5 - h)² + (-2 - k)² = r² (5 - h)² + (2 + k)² = r²
-
For point G(11, -8): (11 - h)² + ((-8) - k)² = r² (11 - h)² + (8 + k)² = r²
Now we have three equations:
- Equation 1: (1 + h)² + (8 + k)² = r²
- Equation 2: (5 - h)² + (2 + k)² = r²
- Equation 3: (11 - h)² + (8 + k)² = r²
Our next step is to eliminate r² from these equations to solve for h and k. A clever way to do this is to set Equation 1 equal to Equation 2 and Equation 1 equal to Equation 3. This will give us two equations with just h and k, making the system solvable.
By equating these expressions, we're essentially saying that the distance from the center (h, k) to each of these points is the same (which, of course, it must be, since that distance is the radius). This is a fundamental property of circles, and leveraging it allows us to simplify the problem significantly.
Remember, setting up the equations correctly is half the battle. Make sure you double-check your substitutions and signs. A small mistake here can propagate through the rest of the solution, leading to an incorrect answer. Accuracy and attention to detail are key!
Solving for h and k
Let's equate Equation 1 and Equation 3 first, because they both have the (8 + k)² term, which will simplify things nicely:
(1 + h)² + (8 + k)² = (11 - h)² + (8 + k)²
Notice that the (8 + k)² terms cancel out on both sides, leaving us with:
(1 + h)² = (11 - h)²
Expanding both sides, we get:
1 + 2h + h² = 121 - 22h + h²
The h² terms also cancel out, simplifying to:
1 + 2h = 121 - 22h
Now, let's solve for h:
24h = 120 h = 5
Great! We found that h = 5. Now, let's use Equation 1 and Equation 2 to solve for k. We have:
(1 + h)² + (8 + k)² = (5 - h)² + (2 + k)²
Substitute h = 5:
(1 + 5)² + (8 + k)² = (5 - 5)² + (2 + k)²
36 + (8 + k)² = 0 + (2 + k)²
Expanding, we get:
36 + 64 + 16k + k² = 4 + 4k + k²
The k² terms cancel out, and we're left with:
100 + 16k = 4 + 4k
Solving for k:
12k = -96 k = -8
So, we have found that h = 5 and k = -8. This means the center of our circle is (5, -8).
Remember, double-check your algebra as you go. Simple mistakes can lead to incorrect values for h and k, which will throw off the rest of the problem. Always be meticulous and review your steps to ensure accuracy!
Finding the Radius (r)
Now that we know the center of the circle is (5, -8), we can find the radius by plugging the center and any of the given points into the circle's equation. Let's use point E(-1, -8):
(x - h)² + (y - k)² = r²
((-1) - 5)² + ((-8) - (-8))² = r²
(-6)² + (0)² = r²
36 = r²
r = √36 r = 6
So, the radius of the circle is 6.
Alternatively, we could have used point F(5, -2) or G(11, -8) to find the radius. Using point F(5, -2) would give us:
(5 - 5)² + (-2 - (-8))² = r²
0² + (6)² = r²
36 = r² r = 6
Using point G(11, -8) would give us:
(11 - 5)² + (-8 - (-8))² = r²
(6)² + (0)² = r²
36 = r² r = 6
As you can see, regardless of which point we choose, the radius remains the same, which confirms our calculations are correct. This consistency is a good sign that we're on the right track!
Writing the Equation and Graphing
Now that we have the center (h, k) = (5, -8) and the radius r = 6, we can write the equation of the circle:
(x - 5)² + (y + 8)² = 36
This is the equation of the circle that passes through the points E(-1, -8), F(5, -2), and G(11, -8).
To graph this circle:
- Plot the center (5, -8) on the coordinate plane.
- From the center, measure out a distance of 6 units in all directions (up, down, left, and right).
- Draw a smooth circle connecting these points.
Your circle should pass through the points E(-1, -8), F(5, -2), and G(11, -8). If it doesn't, double-check your calculations and your graph.
Graphing the circle helps to visualize the solution and ensure that it makes sense. It's a great way to confirm that your equation is correct and that the circle indeed passes through the given points. A visual check can often catch errors that might be missed in the algebraic calculations.
Conclusion
Alright, guys! We successfully found the equation and described how to graph the circle that passes through the points E(-1, -8), F(5, -2), and G(11, -8). The equation is (x - 5)² + (y + 8)² = 36, and the center of the circle is (5, -8) with a radius of 6.
Remember, the key to solving these types of problems is a solid understanding of the standard circle equation, careful substitution, and accurate algebraic manipulation. Don't be afraid to double-check your work and use graphing as a tool to verify your results.
Understanding the concepts thoroughly and practicing similar problems will make you more confident in tackling coordinate geometry questions. Keep honing your skills, and you'll become a master of circles and equations in no time!