Finding Equations Of Lines Through (2, 3) At 45 Degrees To 3x + Y - 8 = 0
This article delves into the process of determining the equations of straight lines that pass through a specific point, (2, 3), and form a 45-degree angle with a given line, 3x + y - 8 = 0. This problem combines concepts from coordinate geometry, including the slope-intercept form of a line, the angle between two lines, and the point-slope form. Understanding these concepts is crucial for solving this type of problem.
Understanding the Fundamentals
Before we dive into the solution, let's refresh the core concepts that underpin this problem:
- Slope-intercept form: A straight line's equation can be expressed as y = mx + c, where 'm' denotes the slope and 'c' represents the y-intercept.
- Slope of a line: The slope (m) of a line signifies its steepness and direction. It is calculated as the change in y divided by the change in x (rise over run).
- Angle between two lines: The angle (θ) between two lines with slopes m1 and m2 is given by the formula: tan θ = |(m1 - m2) / (1 + m1m2)|. This formula is derived from trigonometric principles and helps relate the slopes of lines to the angle they form.
- Point-slope form: The equation of a line passing through a point (x1, y1) with slope m is given by: y - y1 = m(x - x1). This form is particularly useful when a point on the line and its slope are known.
Identifying the Given Information
In this problem, we are given the following:
- A point: (2, 3) through which the required lines must pass.
- A line: 3x + y - 8 = 0, which the required lines make a 45-degree angle with.
- An angle: 45 degrees, which is the angle between the required lines and the given line.
Transforming the Given Line Equation
To determine the slope of the given line, 3x + y - 8 = 0, we can rewrite it in slope-intercept form (y = mx + c). Rearranging the equation, we get:
y = -3x + 8
From this, we can identify the slope of the given line as m1 = -3. This slope is a crucial component in calculating the slopes of the required lines.
Finding the Slopes of the Required Lines
Let m2 be the slope of the required lines. We know that the angle between the given line and the required lines is 45 degrees. Therefore, we can use the formula for the angle between two lines:
tan 45° = |(m2 - m1) / (1 + m1m2)|
Since tan 45° = 1, we have:
1 = |(m2 - (-3)) / (1 + (-3)m2)|
1 = |(m2 + 3) / (1 - 3m2)|
This absolute value equation gives us two possible cases:
Case 1: (m2 + 3) / (1 - 3m2) = 1
Case 2: (m2 + 3) / (1 - 3m2) = -1
Solving for m2 in Case 1
For Case 1, we have:
m2 + 3 = 1 - 3m2
4m2 = -2
m2 = -1/2
This gives us one possible slope for the required line.
Solving for m2 in Case 2
For Case 2, we have:
m2 + 3 = -1 + 3m2
2m2 = 4
m2 = 2
This gives us another possible slope for the required line. Therefore, we have two possible slopes for the lines that meet the given conditions: -1/2 and 2. These slopes, along with the given point, will allow us to determine the equations of the lines.
Determining the Equations of the Lines
Now that we have the slopes of the two possible lines and the point (2, 3) through which they pass, we can use the point-slope form of a line to find their equations.
The point-slope form is given by:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
Equation of the Line with Slope -1/2
Using the point (2, 3) and the slope m2 = -1/2, we get:
y - 3 = (-1/2)(x - 2)
Multiplying both sides by 2 to eliminate the fraction:
2(y - 3) = -1(x - 2)
2y - 6 = -x + 2
Rearranging the equation to the standard form (Ax + By + C = 0):
x + 2y - 8 = 0
This is the equation of the first line that satisfies the given conditions. This line passes through the point (2, 3) and forms a 45-degree angle with the line 3x + y - 8 = 0.
Equation of the Line with Slope 2
Using the point (2, 3) and the slope m2 = 2, we get:
y - 3 = 2(x - 2)
y - 3 = 2x - 4
Rearranging the equation to the standard form (Ax + By + C = 0):
2x - y - 1 = 0
This is the equation of the second line that satisfies the given conditions. This line also passes through the point (2, 3) and forms a 45-degree angle with the line 3x + y - 8 = 0. Thus, we have found both lines that meet the requirements of the problem.
Final Answer
Therefore, the equations of the straight lines passing through the point (2, 3) and making an angle of 45 degrees with the line 3x + y - 8 = 0 are:
- x + 2y - 8 = 0
- 2x - y - 1 = 0
These two lines represent the complete solution to the problem. Each line fulfills the specified conditions, providing a comprehensive answer that demonstrates the application of coordinate geometry principles.
Conclusion
In conclusion, to find the equations of straight lines passing through a given point and making a specific angle with another line, it is essential to utilize the concepts of slope, the angle between lines, and the point-slope form of a line. By systematically applying these principles, we can derive the required equations. This problem underscores the interconnectedness of various concepts in coordinate geometry and highlights the importance of a strong foundation in these principles for problem-solving.
The problem-solving approach involves first identifying the given information, transforming the given line equation to find its slope, and then using the angle formula to find the slopes of the required lines. Finally, the point-slope form is used to determine the equations of the lines. This step-by-step method ensures accuracy and clarity in the solution. Through this detailed explanation, the reader can gain a deeper understanding of the process and apply it to similar problems in the future.
FAQs
Q1: What is the point-slope form of a line and why is it important in solving this problem? The point-slope form of a line is a method to define a line's equation using a single point on the line and the line's slope. The formula is expressed as: y - y1 = m(x - x1), where (x1, y1) are the coordinates of the given point, and m is the slope of the line. This form is particularly useful in solving this problem because we are given a point (2, 3) through which the lines must pass. Once we calculate the slopes of the lines, we can directly substitute the point's coordinates and the slope into the point-slope form to find the equation of each line. This method avoids the need to calculate the y-intercept, making it a straightforward approach for this type of problem.
Q2: Can there be more than two lines that satisfy the given conditions? No, in this specific scenario, there can only be two distinct lines that satisfy the conditions. The reason lies in the nature of angles and slopes. When a line intersects another line at a given angle (in this case, 45 degrees), there are two possible directions the intersecting line can take, one on each side of the original line. Each direction corresponds to a unique slope. Since we have calculated two distinct slopes that satisfy the angle condition, and each slope, combined with the given point, defines a unique line, there can only be two lines that meet all the specified requirements. This is a fundamental property of straight lines and angles in Euclidean geometry.
Q3: How does the angle between two lines relate to their slopes? The angle between two lines is intimately related to their slopes through a trigonometric formula. If two lines have slopes m1 and m2, and the angle between them is θ, then the relationship is given by: tan θ = |(m1 - m2) / (1 + m1m2)|. This formula is derived from the principles of trigonometry and coordinate geometry and allows us to calculate the angle between two lines if we know their slopes, or conversely, to find the slope of a line if we know the slope of another line and the angle between them. In this problem, we used this formula to find the slopes of the lines that make a 45-degree angle with the given line. Understanding this relationship is crucial for solving problems involving angles between lines.
Q4: What are some common mistakes to avoid when solving this type of problem? There are several common mistakes that students often make when solving problems involving lines and angles. One frequent error is forgetting the absolute value in the formula for the angle between two lines: tan θ = |(m1 - m2) / (1 + m1m2)|. The absolute value is crucial because the angle can be measured in two directions, leading to two possible slopes. Another mistake is incorrectly manipulating the algebraic equations when solving for the slopes. It is important to carefully follow the steps and avoid arithmetic errors. Additionally, some students may forget to consider both cases arising from the absolute value, resulting in only one solution instead of two. Finally, a lack of understanding of the fundamental concepts, such as the point-slope form and the meaning of slope, can lead to incorrect solutions. To avoid these mistakes, it is essential to practice similar problems and thoroughly understand the underlying principles.
