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. \( y + x = 5 \)
• E. \( 2y + x = 5 \)

Hint:

When two lines are perpendicular, their slopes are negative reciprocals of each other. Use this property to find the equation of the perpendicular line.

Answer 1

The Correct Option is A

Step 1: Find the slope of line "l".
The slope \( m_1 \) of line "l" passing through points \( (4,1) \) and \( (8,-1) \) is given by: \[ m_1 = \frac{-1 - 1}{8 - 4} = \frac{-2}{4} = -\frac{1}{2} \] Step 2: Find the slope of line "k".
Since lines "l" and "k" are perpendicular, the slope of line "k", denoted \( m_2 \), is the negative reciprocal of \( m_1 \): \[ m_2 = \frac{2}{1} = 2 \] Step 3: Use the point-slope form of the equation of a line.
The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] Substitute the point \( (3,1) \) and slope \( 2 \) into the equation: \[ y - 1 = 2(x - 3) \] Simplify: \[ y - 1 = 2x - 6 \quad \Rightarrow \quad 2x - y = 5 \] Step 4: Conclusion.
The correct answer is (A).