Finding Circle And Line Intersection In Quadrant I
Hey guys! Let's dive into a cool math problem. We're gonna find out where a line and a circle meet, specifically in the first quadrant. This kind of problem is super common in math and has real-world applications, too. Think about it: GPS systems, game development, and even architecture use these principles. So, understanding how lines and circles interact is a valuable skill.
The Big Picture: We've got a line represented by the equation y = 2.5x + 2. This equation tells us the relationship between the x and y coordinates of every point on the line. Then, we have a circle with a radius of 5, centered at the point (0, 2). This means every point on the circle's edge is exactly 5 units away from the center. Our mission? Figure out the exact spot (or spots) where the line crosses the circle, but only in the first quadrant. Remember, the first quadrant is where both x and y values are positive. This problem is a blend of algebra and geometry, so buckle up! We'll use equations to pinpoint the exact location of the intersection. We will start with a general understanding of the circle and the line.
Understanding the Circle and the Line
First things first, let's break down the components. The line equation y = 2.5x + 2 is in slope-intercept form (y = mx + b), where m is the slope, and b is the y-intercept. In our case, the slope (m) is 2.5, which means for every 1 unit increase in x, the y value increases by 2.5 units. The y-intercept (b) is 2, indicating the line crosses the y-axis at the point (0, 2). Think of it as the line's starting point. The circle, defined by its center (0, 2) and radius 5, is described by the standard equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center, and r is the radius. Substituting the given values, we get (x - 0)^2 + (y - 2)^2 = 5^2, which simplifies to x^2 + (y - 2)^2 = 25. This equation tells us all the points (x, y) that make up the circle. Now we have a clear definition of the line and the circle.
We know where the center of the circle is, so it makes it easier to work with. The line intersects the y-axis at the center of the circle. We're looking for where the line and circle intersect, meaning where their coordinates (x, y) are the same. Visually, imagine drawing the line and the circle on a graph. The points where the line crosses the circle are the solutions to our problem. We will use the equations and do some algebraic manipulation to find the intersection points.
Setting Up the Equations for Intersection
To find where the line and circle intersect, we need to solve the system of equations. We have:
y = 2.5x + 2(the line)x^2 + (y - 2)^2 = 25(the circle)
Since we know y in terms of x from the line equation, we can substitute 2.5x + 2 for y in the circle equation. This will give us a new equation with only x as the variable. Doing this substitution, we get x^2 + (2.5x + 2 - 2)^2 = 25, which simplifies to x^2 + (2.5x)^2 = 25. Now we're cookin'! This is a quadratic equation, which we can solve to find the x-coordinates of the intersection points. Remember that quadratic equations can have zero, one, or two solutions. This will depend on if the line intersects the circle. After doing some algebra, we will get the intersection points. Let's do a quick recap. We have a line and a circle, and our main goal is to find their intersection points. The rest is solving the equations to find the coordinates of these intersection points.
Solving the Quadratic Equation
Let's keep going! Simplifying our equation x^2 + (2.5x)^2 = 25, we get x^2 + 6.25x^2 = 25, which combines to 7.25x^2 = 25. To isolate x squared, divide both sides by 7.25: x^2 = 25 / 7.25. This gives us x^2 ≈ 3.448. To find x, we take the square root of both sides: x ≈ ±√3.448. This results in two possible values for x: approximately 1.857 and -1.857. We're almost there! We've found the x-coordinates of the intersection points. Now, we'll plug these x-values back into the line equation y = 2.5x + 2 to find the corresponding y-coordinates. For x ≈ 1.857, we have y ≈ 2.5 * 1.857 + 2 ≈ 6.643. So, one intersection point is approximately (1.857, 6.643). For x ≈ -1.857, we have y ≈ 2.5 * -1.857 + 2 ≈ -2.643. This gives us another point: approximately (-1.857, -2.643). Remember, we're looking for the intersection in the first quadrant where both x and y are positive. So, only the first solution is valid. The second solution is in the third quadrant, so it is invalid. Let's keep moving.
Finding the Intersection Point in the First Quadrant
We calculated two potential intersection points: (1.857, 6.643) and (-1.857, -2.643). Now, we must check which of these points lies in the first quadrant, where both x and y are positive. The first point, (1.857, 6.643), has positive x and y values. This means it lies in the first quadrant. The second point, (-1.857, -2.643), has negative x and y values, placing it in the third quadrant. Therefore, it is not a solution to our problem. So, the only intersection point in the first quadrant is approximately (1.857, 6.643). When stating the answer to 3 decimal places, we get (1.857, 6.643). Now we know the exact point where the line and circle intersect in the first quadrant. To get a better understanding of the answer, we can make a graph. We can plot the line, plot the circle, and see the intersection points. Now, we are done with the problem.
Final Answer and Conclusion
So, the line with the equation y = 2.5x + 2 intersects the circle with a radius of 5 and center (0, 2) at approximately (1.857, 6.643) in the first quadrant. Great job, guys! We successfully navigated this math problem by using equations for the line and circle, substituting, and solving for the intersection points. This approach can be used in various applications. Remember, the key is understanding the equations, doing the algebra, and knowing your quadrants. This problem reinforces the relationship between algebra and geometry, showing how mathematical tools can solve real-world problems. Keep practicing and exploring – you'll be amazed at what you can figure out!