In What Ratio Is The Line Segment Joining The Points (- 3 2 And 6'1 Is Divided By Y Axis?

In What Ratio Is The Line Segment Joining The Points (-3, 2) And (6, 1) Divided By Y Axis?

Understanding how a line segment interacts with the coordinate axes is fundamental in coordinate geometry. When dealing with points plotted on the Cartesian plane, a common question involves determining the point where a line segment connecting two points intersects a particular axis, and then finding the ratio in which this point divides the segment. Specifically, the problem: In what ratio is the line segment joining the points (-3, 2) and (6, 1) divided by the y-axis? is a classic example that helps deepen one's understanding of coordinate geometry concepts like section formula, division ratios, and the intersection of lines with axes.

This comprehensive article aims to explore this problem step by step, providing clarity on the methods used to find the ratio and explaining the underlying principles involved. Whether you're a student preparing for exams or an enthusiast seeking to solidify your understanding of coordinate geometry, this guide will serve as an informative resource.

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Understanding the Problem

Before diving into calculations, it's essential to interpret what the problem is asking.

• Points involved:


• Point A: (-3, 2)


• Point B: (6, 1)


• Line segment: The straight line connecting A and B.


• Axis of interest: The y-axis, which is the line x = 0.


• Question: The line segment AB is intersected by the y-axis at a certain point, say P. The goal is to find the ratio in which P divides the segment AB.

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Fundamental Concepts in Coordinate Geometry

To effectively solve this problem, it’s crucial to understand some key concepts:

1. Coordinates of Points

Every point in a plane is represented by an ordered pair (x, y), where x and y are real numbers.

2. The Line Segment Connecting Two Points

Given points A(x₁, y₁) and B(x₂, y₂), the segment AB is the straight line between these points.

3. Intersection with the Y-Axis (x = 0)

Any point on the y-axis has an x-coordinate of zero. To find where the segment AB intersects the y-axis, we need to find the point P(x=0, y).

4. Section Formula

The section formula gives the coordinates of a point dividing a line segment in a given ratio.

• If a point P divides the segment AB in the ratio m:n, then:

\[ xP = \frac{mx2 + nx1}{m + n} \quad \text{and} \quad yP = \frac{my2 + ny1}{m + n} \]

• Conversely, if we know the coordinates of P and the points A and B, we can find the ratio m:n.

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Step-by-Step Solution

Now, we'll proceed to find the point of intersection P of the segment AB with the y-axis, and then determine the ratio in which P divides AB.

Step 1: Write down the coordinates of the points

• A: (-3, 2)
• B: (6, 1)

Step 2: Find the coordinates of the intersection point P with x=0

The point P lies on the line segment AB and on the y-axis, which has x=0.

The equation of the line passing through A and B can be written parametrically or in slope-intercept form.

Step 3: Find the equation of line AB

• Slope of AB:

\[ m = \frac{y2 - y1}{x2 - x1} = \frac{1 - 2}{6 - (-3)} = \frac{-1}{9} = -\frac{1}{9} \]

• Equation of line AB:

Using point-slope form with point A(-3, 2):

\[
y - 2 = -\frac{1}{9}(x + 3)
\]

Simplify:

\[
y - 2 = -\frac{1}{9}x - \frac{1}{3}
\]
\[
y = -\frac{1}{9}x - \frac{1}{3} + 2
\]
\[
y = -\frac{1}{9}x + \frac{5}{3}
\]

Step 4: Find the intersection point P (x=0)

Set x=0 in the line equation:

\[
y = -\frac{1}{9}(0) + \frac{5}{3} = \frac{5}{3}
\]

Thus, the point of intersection is:

\[
P = (0, \frac{5}{3})
\]

Determining the Division Ratio

Now, with P(0, 5/3), and points A(-3, 2), B(6, 1), we need to find the ratio m:n in which P divides AB.

Step 5: Apply the section formula in reverse

• We know P divides AB in ratio m:n, and:

\[ xP = \frac{mx2 + nx_1}{m + n} \] \[ yP = \frac{my2 + ny_1}{m + n} \]

• Use the x-coordinate:

\[ 0 = \frac{m \times 6 + n \times (-3)}{m + n} \]

Multiply both sides by (m + n):

\[
0 = 6m - 3n
\]

Rearranged:

\[
6m = 3n \Rightarrow 2m = n
\]

• Use the y-coordinate to find the actual ratio:

\[
\frac{5}{3} = \frac{m \times 1 + n \times 2}{m + n}
\]

Substitute n = 2m:

\[
\frac{5}{3} = \frac{m \times 1 + 2m \times 2}{m + 2m} = \frac{m + 4m}{3m} = \frac{5m}{3m} = \frac{5}{3}
\]

This confirms the ratio is consistent.

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Final Answer and Interpretation

From the calculations:

\[
n = 2m
\]

Dividing both sides by m:

\[
\frac{n}{m} = 2
\]

Therefore, the point P divides the segment AB in the ratio:

\[
\boxed{
\text{Ratio } \frac{AP}{PB} = \frac{m}{n} = \frac{1}{2}
}
\]

Alternatively, the segment is divided in the ratio 1:2, with:

• AP : PB = 1 : 2

This means that the point P, where the segment AB intersects the y-axis, divides the segment into two parts, with AP being one-third and PB being two-thirds of the total length of AB.

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Summary of Key Steps

1. Identify points and axes: Recognize the points involved and the line of intersection (y-axis, x=0).
2. Find the equation of the line AB: Calculate the slope and derive the line's equation.
3. Calculate the intersection point: Plug x=0 into the line equation to find P.
4. Apply the section formula: Use the coordinates of A, B, and P to find the ratio m:n.
5. Interpret the ratio: Understand what the ratio indicates about how P divides AB.

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Additional Tips and Concepts

• Line equations: Always check the slope before writing the line equation.
• Intersection points: To find where a line intersects an axis, substitute the axis coordinate (x=0 or y=0).
• Section formula: Useful for dividing a segment internally in a given ratio.
• Ratios and distances: The ratio m:n can also be interpreted in terms of distances from P to A and B, especially when considering the division of the segment.

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Practical Applications

Understanding how to find the division ratio of a segment by a line is important in various fields such as:

• Engineering: Structural analysis and design.


• Computer Graphics: Rendering and object positioning.


• Navigation: Dividing routes or paths proportionally.


• Mathematics Education: Building foundational knowledge in coordinate geometry.

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Conclusion

In conclusion, the line segment joining the points (-3, 2) and (6, 1) is divided by the y-axis at a point that splits the segment in a ratio of 1:2. This detailed exploration demonstrates the application of the slope-intercept form, intersection calculation, and the section formula to solve real-world problems in coordinate geometry. Mastering these techniques enhances problem-solving skills and provides a strong foundation for more advanced mathematical concepts.

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Keywords: coordinate geometry, division ratio, line segment, intersection with y-axis, section formula, points, slope, equations of lines, mathematical problem-solving