Finding Intersection: Line And Circle In Quadrant One
Hey guys! Let's dive into a cool math problem. We're going to figure out where a line and a circle meet in the first quadrant of the coordinate plane. This is a common type of problem in geometry, and it's super useful for understanding how different equations and shapes interact. We'll be using some algebra and a little bit of visualization to nail this down. So, let's get started. The first thing we need to do is break down what we're working with: a line represented by the equation $y = x + 5$, and a circle, which has a radius of 5 and is centered at the point $(0, 5)$. The heart of this question is finding the points where the line and the circle cross paths.
Before we jump into the math, it's helpful to visualize what's going on. Imagine the x-y coordinate plane. The first quadrant is the top-right section where both x and y values are positive. The line $y = x + 5$ is a straight line that slopes upwards. The "+ 5" tells us where it crosses the y-axis (at the point (0, 5)). The circle with a radius of 5 and centered at (0, 5) is a shape that will intersect with the y-axis, and because the center is at (0,5), it will touch the x-axis, the maximum will be at x=5. Understanding the shapes involved is essential for solving the problem. So, we're looking for where these two shapes touch within the positive x and y values. Also, remember that a circle is defined by its center and radius, so knowing these two properties will allow us to represent it in the coordinate plane. The standard form of a circle's equation is: $(x - h)^2 + (y - k)^2 = r^2$, where (h, k) is the center of the circle, and r is the radius. We'll use this to get our circle's equation. This is the foundation upon which we will build our solution. Remember that the first quadrant is like a playground for positive numbers, so the solution, if there is one, must exist there.
Setting up the Equations
Now, let's get down to business and use our knowledge to formulate a plan. The first step involves converting the word problem into mathematical equations so that we can solve it systematically. We have two core pieces of information, and we will translate this information into equations, which we can then solve. First, we know the line's equation is: $y = x + 5$. This is a nice, straightforward linear equation. Second, we know the circle's center is at (0, 5) and its radius is 5. Using the standard form of a circle's equation $(x - h)^2 + (y - k)^2 = r^2$, where (h, k) is the center and r is the radius, we can plug in the values to get the equation for our circle: $(x - 0)^2 + (y - 5)^2 = 5^2$. Simplifying this, we get: $x^2 + (y - 5)^2 = 25$. These are the equations that will help us find our answer. The line's equation defines all the points that fall on the line, and the circle's equation defines all the points that lie on the circumference of the circle. To find where they intersect, we need to find the points that satisfy both equations. This is the essence of solving simultaneous equations - finding the values of x and y that work for both the line and the circle. The next stage of the journey involves solving the equations.
Solving for the Intersection Points
Alright, it's time to find those intersection points. Because we know that at the intersection, both equations must be true, we can use substitution. The line equation is $y = x + 5$, which gives us a straightforward way to substitute y in the circle's equation. Remember that the x and y values at the intersection point satisfy both equations. Now, take the circle's equation: $x^2 + (y - 5)^2 = 25$, and replace 'y' with 'x + 5' (because that's what 'y' equals according to the line equation). This gives us: $x^2 + ((x + 5) - 5)^2 = 25$. Let's simplify that: $x^2 + (x)^2 = 25$. And then: $2x^2 = 25$. Now, solve for x: $x^2 = rac25}{2}$, so $x = ± rac{5}{\sqrt{2}}$. Now that we have the x-values, we can find the corresponding y-values using the line's equation, $y = x + 5$. For $x = rac{5}{\sqrt{2}}$, $y = \frac{5}{\sqrt{2}} + 5$. For $x = - rac{5}{\sqrt{2}}$, $y = - rac{5}{\sqrt{2}} + 5$. We now have two possible points{\sqrt{2}}, \frac{5}{\sqrt{2}} + 5\right)$ and $\left(-\frac{5}{\sqrt{2}}, -\frac{5}{\sqrt{2}} + 5\right)$. But wait, remember the first quadrant? In the first quadrant, both x and y must be positive. Therefore, only the point $\left(\frac{5}{\sqrt{2}}, \frac{5}{\sqrt{2}} + 5\right)$ fits the bill. Now we have found our answer. Remember, the goal of this exercise is to build your ability to solve complex problems.
Analyzing the Solution and the First Quadrant
So, what does our solution tell us? First, let's be sure we're on the same page and fully understand the coordinate plane. Remember that the first quadrant is a region where both x and y coordinates are positive. This is where our line and circle meet according to the question. When we calculated the intersection points, we got two possible solutions, but only one of these fell in the first quadrant. To recap, we found that the line $y = x + 5$ intersects the circle with center (0, 5) and radius 5 at the point $\left(\frac{5}{\sqrt{2}}, \frac{5}{\sqrt{2}} + 5\right)$. Let's break down this point: The x-coordinate is approximately 3.54, which is positive. The y-coordinate is approximately 8.54, also positive. Therefore, the point is indeed in the first quadrant. This confirms our understanding of the coordinate system. If you're wondering, the second point we calculated (with negative x and y) would fall somewhere in the third quadrant (where both x and y are negative). Let's take a look. If we were to plot the line and the circle on a graph, we would visually see the intersection point in the first quadrant. This visualization is a crucial part of understanding. In this scenario, we only had one point of intersection in the first quadrant. Remember, always consider the context of the problem. Knowing the limitations and boundaries will allow you to quickly discard invalid results and save yourself a lot of time and effort.
Conclusion: The Intersection Point
Great job, guys! We've successfully found the point where the line and the circle intersect in the first quadrant. The key to this problem was using the equations for both the line and the circle, then using a method (substitution) to solve the simultaneous equations. Remember, the intersection point must satisfy both equations. We used the line equation to substitute y in the circle equation. The solution: The line $y = x + 5$ intersects the circle with radius 5 and center (0,5) in the first quadrant at the point $\left(\frac{5}{\sqrt{2}}, \frac{5}{\sqrt{2}} + 5\right)$. This means that at this specific point, the line and the circle share the same x and y coordinates. This point lies approximately at (3.54, 8.54). Remember, practice is key. Try similar problems with different equations and see how you can apply these steps. This is a solid foundation for more complex geometry problems. Keep experimenting with different equations and shapes, and you'll become more confident in your problem-solving skills. Feel free to ask if you have more questions. Keep up the awesome work!
