If The Line Passes Through The Points (-8, R) And(2,-10) Is Perpendicular To The Line 2x - Y = 6.find

If The Line Passes Through The Points (-8, R) And (2, -10) Is Perpendicular To The Line 2x - Y = 6.find

Understanding how to determine the characteristics of a line passing through specific points and its relationship to other lines is fundamental in coordinate geometry. When given points with variables and the requirement to identify perpendicularly intersecting lines, it involves calculating slopes, equations, and verifying perpendicularity conditions. This comprehensive guide will walk you through the process of solving such problems step-by-step, ensuring you grasp the concepts clearly and can apply them to similar questions.

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Understanding the Problem Statement

Before diving into calculations, it’s essential to interpret what the problem is asking:

• You are given two points: (-8, R) and (2, -10). Here, R is an unknown y-coordinate.


• You need to determine the value of R such that the line passing through these points is perpendicular to another line.


• The other line is given by the equation 2x - y = 6.


• Your goal is to find the value of R that makes the line through the points perpendicular to the given line.

This requires knowledge of:

• Calculating the slope of a line given two points.


• Finding the slope of the given line.


• Understanding the concept of perpendicular slopes.


• Using these to solve for R.

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Key Concepts in Coordinate Geometry

1. Slope of a Line

The slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated as:

\[ m = \frac{y2 - y1}{x2 - x1} \]

This measure indicates the steepness of the line and its direction.

2. Equation of a Line

The slope-intercept form is:

\[ y = mx + c \]

where:

• m is the slope,


• c is the y-intercept.

3. Perpendicular Lines

Two lines are perpendicular if the product of their slopes is -1:

\[ m1 \times m2 = -1 \]

This property is crucial for solving the problem, as it allows you to find the unknown slope (and R) by using the slopes’ relation.

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Step-by-Step Solution Approach

Let's break down the problem into manageable steps:

Step 1: Find the slope of the given line 2x - y = 6

Rewrite the line in slope-intercept form:

\[ 2x - y = 6 \]
\[ -y = -2x + 6 \]
\[ y = 2x - 6 \]

Thus, the slope of the given line is:

\[ m_{line} = 2 \]

Step 2: Determine the slope of the line passing through the points (-8, R) and (2, -10)

Let’s denote this slope as \( m_{AB} \):

\[ m_{AB} = \frac{-10 - R}{2 - (-8)} = \frac{-10 - R}{10} \]

Step 3: Use the perpendicularity condition

Since the line passing through the points is perpendicular to the given line, their slopes satisfy:

\[ m{AB} \times m{line} = -1 \]

Substitute the known slope \( m_{line} = 2 \):

\[ \left( \frac{-10 - R}{10} \right) \times 2 = -1 \]

Simplify:

\[ \frac{-10 - R}{10} \times 2 = -1 \]
\[ \frac{2(-10 - R)}{10} = -1 \]
\[ \frac{-20 - 2R}{10} = -1 \]

Multiply both sides by 10 to clear the denominator:

\[ -20 - 2R = -10 \]

Step 4: Solve for R

Rearranged:

\[ -2R = -10 + 20 \]
\[ -2R = 10 \]
\[ R = -\frac{10}{2} \]
\[ R = -5 \]

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Final Answer and Explanation

The value of R that makes the line passing through (-8, R) and (2, -10) perpendicular to the line 2x - y = 6 is R = -5.

This result signifies that when the y-coordinate of the first point is -5, the line connecting it to the point (2, -10) is perpendicular to the given line.

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Additional Insights and Related Concepts

1. Verifying the Result

It's good practice to verify the solution:

• Calculate the slope of the line passing through (-8, -5) and (2, -10):

\[ m_{AB} = \frac{-10 - (-5)}{2 - (-8)} = \frac{-10 + 5}{10} = \frac{-5}{10} = -\frac{1}{2} \]

• Confirm perpendicularity:

\[ m{AB} \times m{line} = -\frac{1}{2} \times 2 = -1 \]

which confirms the correctness.

2. Practical Applications

Understanding these concepts is vital in various fields such as:

• Engineering design
• Computer graphics
• Physics for analyzing trajectories
• Data analysis for trend lines

3. Common Mistakes to Avoid

• Forgetting to convert equations into slope-intercept form
• Incorrectly calculating the slope
• Mixing up the order of points when computing the slope
• Not applying the perpendicularity condition correctly

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Tips for Solving Similar Problems

• Always write the line equations in slope-intercept form for clarity.
• Carefully handle the signs during calculations.
• Use the property \( m1 \times m2 = -1 \) for perpendicular lines.
• Double-check your algebraic manipulations.

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Conclusion

In summary, when you need to find the value of R such that a line passing through points (-8, R) and (2, -10) is perpendicular to a given line like 2x - y = 6, the process involves:

• Finding the slope of the given line.


• Establishing the slope of the passing line in terms of R.


• Applying the perpendicularity condition to set up an equation.


• Solving for R.

This problem demonstrates the power of coordinate geometry principles in solving real-world and academic problems efficiently. Mastering these steps enhances your ability to approach similar questions confidently and accurately.

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