Finding The Circle: Radius, Center, And Equation Explained

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Hey everyone! Today, we're diving into a cool geometry problem: figuring out the equation of a circle. This particular circle has a radius of 3, and its center is located at the point where two lines – 2x + 3y = 5 and 2x - y = 1 – intersect. Sounds fun, right? Don't worry, it's not as scary as it sounds. We'll break it down step by step, so even if you're new to this, you'll be able to follow along. We'll start with finding the center of the circle, then we’ll use that and the radius to determine the circle's equation. Let's get started, shall we?

Unveiling the Circle's Center: Intersection of Lines

First things first, we need to find the center of our circle. This center point is where the two lines, 2x + 3y = 5 and 2x - y = 1, cross each other. This is a classic algebra problem involving solving a system of linear equations. There are several methods to tackle this, but we'll use the elimination method here. It's often the quickest way to get the solution. The core idea is to manipulate the equations so that when we add or subtract them, one of the variables (either x or y) disappears. We're left with a single-variable equation that we can easily solve.

Let’s label our equations:

Equation 1: 2x + 3y = 5

Equation 2: 2x - y = 1

Notice that both equations have 2x. This is great because if we subtract Equation 2 from Equation 1, the x terms will cancel out. Let's do that:

(2x + 3y) - (2x - y) = 5 - 1

Simplifying this, we get:

2x - 2x + 3y + y = 4

4y = 4

Now, to solve for y, we divide both sides by 4:

y = 1

Awesome! We've found the y-coordinate of the center, which is 1. Now, we'll plug this value of y back into either Equation 1 or Equation 2 to find the value of x. Let's use Equation 2 because it looks a bit simpler:

2x - y = 1

Substitute y = 1:

2x - 1 = 1

Add 1 to both sides:

2x = 2

Divide both sides by 2:

x = 1

So, the x-coordinate of the center is also 1. Therefore, the center of the circle is the point (1, 1). We've successfully found the center. High five, guys!

Constructing the Circle's Equation: Using Radius and Center

Now that we've found the center of the circle (1, 1) and we know the radius is 3, we can build the equation of the circle. The standard form equation for a circle is a really handy formula that relates the x and y coordinates of any point on the circle to its center and radius. This formula is:

(x - h)² + (y - k)² = r²

Where:

  • (h, k) represents the coordinates of the center of the circle.
  • r represents the radius of the circle.

In our case, the center (h, k) is (1, 1) and the radius r is 3. Let's plug these values into the standard equation:

(x - 1)² + (y - 1)² = 3²

Simplifying further:

(x - 1)² + (y - 1)² = 9

And there you have it! This is the equation of the circle with a radius of 3 and its center at the intersection point of the lines 2x + 3y = 5 and 2x - y = 1. This equation means that if you pick any point (x, y) that satisfies this equation, that point will lie on the circle. Any point that doesn't satisfy the equation won't be on the circle. The power of this equation is that it fully describes the circle in the coordinate plane. You can use it to determine if a given point is inside, outside, or on the circle. You could even use it to graph the circle precisely. Understanding and applying this standard equation is fundamental in analytic geometry and opens the door to solving more complex problems involving circles and other geometric shapes.

Expanding the Equation (Optional)

Sometimes, you might see the equation of a circle written in a slightly different form. You can expand the terms in our equation to get the general form of the circle's equation. This is not always necessary for solving problems, but it's good to know. Let's expand the equation we found earlier: (x - 1)² + (y - 1)² = 9

First, expand (x - 1)². This is the same as (x - 1) * (x - 1). Using the distributive property (or FOIL method), we get:

x² - 2x + 1

Next, expand (y - 1)². This is the same as (y - 1) * (y - 1). Using the distributive property, we get:

y² - 2y + 1

Now, substitute these expanded terms back into the original equation:

x² - 2x + 1 + y² - 2y + 1 = 9

Combine the constant terms (1 + 1 = 2) and move the 9 to the left side by subtracting it from both sides:

x² - 2x + y² - 2y + 2 - 9 = 0

Simplifying this, we get:

x² + y² - 2x - 2y - 7 = 0

This is the general form of the equation of the circle. It's the same circle, just written in a different way. While the standard form (x - h)² + (y - k)² = r² is usually more helpful for identifying the center and radius directly, the general form can be useful in certain types of problems. The general form is also a quadratic equation, which is where things can get more complicated. Being able to go between standard and general forms helps give you a deeper understanding and better grasp of the concept and problem-solving skills.

Wrapping Up and Key Takeaways

And there you have it! We've successfully determined the equation of a circle given its radius and the intersection point of two lines. Let's recap what we did and what we've learned.

  • Finding the Center: We started by finding the center of the circle by solving a system of linear equations using the elimination method. This involved manipulating the equations to eliminate one variable, solving for the other, and then back-substituting to find the remaining variable. This is a fundamental algebra skill that is useful in various mathematical problems.
  • Using the Standard Equation: We used the standard form of the circle equation, (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. We plugged in the values for the center (1, 1) and radius (3) to get the final equation: (x - 1)² + (y - 1)² = 9.
  • Understanding the Equation's Significance: The equation is a compact way to describe the circle. Any point (x, y) that satisfies the equation lies on the circle. The equation fully defines the circle's position and size in the coordinate plane. We also touched upon the general form of the equation and its relationship to the standard form. We expanded our understanding of the equation.

Hopefully, this detailed walkthrough has helped you understand how to find the equation of a circle. Remember, the key is to break the problem down into smaller, manageable steps. Practice is essential, so try working through similar problems on your own to solidify your understanding. Geometry, like any branch of math, becomes easier with practice. Keep exploring, keep learning, and don't be afraid to tackle new challenges. Good luck, and keep up the great work, everyone! We have reached the end of the explanation, so now you can try the practice questions on your own. Remember to review the steps if you get stuck. Keep up the good work!