Question

Lines "l" and "k" are perpendicular to each other. And line "l" passes through points (4,1) and (8, -1). What is the equation of the line "k" which passes through the point (3,1)?

• A. 2y - x = 5
• B. 2x - y = 5
• C. y - 2x = 5
• D. y + 2x = 5
• E. 2y + x = 5

Hint:

Remember the slope relationships: for parallel lines, slopes are equal (\(m_1 = m_2\)); for perpendicular lines, slopes are negative reciprocals (\(m_1 = -1/m_2\)). This is a fundamental concept in coordinate geometry.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves finding the equation of a line given a point and its relationship (perpendicularity) to another line. The key is the relationship between the slopes of perpendicular lines.
Step 2: Detailed Explanation:
1. Find the slope of line l.
The slope (\(m_l\)) of a line passing through points (\(x_1, y_1\)) and (\(x_2, y_2\)) is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the points (4,1) and (8, -1) for line l: \[ m_l = \frac{-1 - 1}{8 - 4} = \frac{-2}{4} = -\frac{1}{2} \] 2. Find the slope of line k.
If two lines are perpendicular, the slope of one is the negative reciprocal of the slope of the other. Let the slope of line k be \(m_k\). \[ m_k = -\frac{1}{m_l} = -\frac{1}{(-1/2)} = 2 \] 3. Find the equation of line k.
We know that line k has a slope of 2 and passes through the point (3,1). We can use the point-slope form of a linear equation: \( y - y_1 = m(x - x_1) \). \[ y - 1 = 2(x - 3) \] \[ y - 1 = 2x - 6 \] Rearrange the equation to match the format of the options: \[ 6 - 1 = 2x - y \] \[ 5 = 2x - y \] This can be written as \(2x - y = 5\).
Step 3: Final Answer:
The equation of line k is \(2x - y = 5\), which corresponds to option (B).