Ratio And Intersection Point Y-axis Division Of Line Segment

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Introduction

In coordinate geometry, a fundamental problem involves finding the ratio in which a line segment is divided by one of the coordinate axes. This often extends to determining the exact point of intersection. In this article, we will explore a specific instance of this problem: finding the ratio in which the y-axis divides the line segment joining the points (4, -5) and (-1, 2), and subsequently, identifying the coordinates of the point of intersection. This exploration will enhance your understanding of section formulas and their applications in coordinate geometry.

Problem Statement

The core of our discussion revolves around the following question: In what ratio does the y-axis divide the line segment that connects the points A(4, -5) and B(-1, 2)? Furthermore, what are the coordinates of the point where this division occurs? This problem is a classic example of how coordinate geometry combines algebraic and geometric concepts. To solve it, we will use the section formula, a cornerstone in coordinate geometry that helps us find the coordinates of a point dividing a line segment in a given ratio.

Understanding the Section Formula

The section formula is a vital tool for solving problems related to the division of line segments. It states that if a point P(x, y) divides the line segment joining two points A(x₁, y₁) and B(x₂, y₂) in the ratio m : n, then the coordinates of point P can be calculated as follows:

  • x = (m x₂ + n x₁) / (m + n)
  • y = (m y₂ + n y₁) / (m + n)

This formula allows us to determine the exact location of the point P based on the coordinates of A and B and the ratio m : n. It is derived from basic principles of similar triangles and coordinate geometry, making it a robust and universally applicable method in various mathematical contexts. Understanding and applying this formula correctly is crucial for solving the problem at hand.

Applying the Section Formula to Our Problem

Now, let's apply the section formula to our specific problem. We are given the points A(4, -5) and B(-1, 2), and we need to find the ratio in which the y-axis divides the line segment AB. Since the dividing point lies on the y-axis, its x-coordinate must be 0. Let's denote the point of intersection as P(0, y) and assume that the y-axis divides the line segment AB in the ratio k : 1. This assumption simplifies our calculations and allows us to solve for the unknown ratio k.

Using the section formula, the coordinates of point P can be expressed as:

  • 0 = (k(-1) + 1(4)) / (k + 1)
  • y = (k(2) + 1(-5)) / (k + 1)

From the first equation, we can solve for k, which will give us the ratio in which the y-axis divides the line segment. The second equation will then help us find the y-coordinate of the point of intersection. This approach highlights the power of the section formula in solving geometric problems algebraically. The process involves setting up the equations correctly and then solving them to find the desired ratio and coordinates.

Solving for the Ratio k

To find the ratio k, we focus on the equation derived from the x-coordinate of point P:

0 = (k(-1) + 1(4)) / (k + 1)

This equation simplifies to:

0 = -k + 4

Solving for k, we get:

k = 4

This result indicates that the y-axis divides the line segment AB in the ratio 4:1. The positive value of k confirms that the point of intersection lies between A and B. This step is crucial as it provides the necessary ratio to determine the exact point of intersection. The algebraic manipulation here is straightforward but essential for arriving at the correct geometric interpretation.

Determining the Point of Intersection

Now that we have found the ratio k = 4, we can use this value to find the y-coordinate of the point of intersection. Recall the equation for the y-coordinate of point P:

y = (k(2) + 1(-5)) / (k + 1)

Substituting k = 4 into the equation, we get:

y = (4(2) + 1(-5)) / (4 + 1) y = (8 - 5) / 5 y = 3 / 5

Therefore, the coordinates of the point of intersection P are (0, 3/5). This result completes the solution, providing not only the ratio in which the y-axis divides the line segment but also the precise location of the intersection point. The process involves a straightforward substitution and arithmetic calculation, reinforcing the practical application of the section formula.

Conclusion

In conclusion, we have successfully determined that the y-axis divides the line segment joining the points (4, -5) and (-1, 2) in the ratio 4:1. Furthermore, the point of intersection is (0, 3/5). This problem exemplifies the utility of the section formula in coordinate geometry, allowing us to solve geometric problems using algebraic methods. The process involved understanding the problem, applying the section formula, solving for the unknown ratio, and finally, determining the coordinates of the point of intersection. This comprehensive approach is valuable for tackling similar problems in coordinate geometry and reinforces the fundamental principles of mathematical problem-solving.

This exploration not only enhances our understanding of the section formula but also highlights the interconnectedness of algebra and geometry. The ability to translate geometric problems into algebraic equations and solve them is a key skill in mathematics. The steps we followed – from setting up the problem using the section formula to solving for the ratio and the point of intersection – provide a structured approach that can be applied to a wide range of problems in coordinate geometry and beyond. By mastering these techniques, students and enthusiasts alike can gain a deeper appreciation for the elegance and power of mathematical tools in solving real-world and theoretical problems.

Final Answer

The y-axis divides the line segment in the ratio 4:1, and the point of intersection is (0, 3/5).