If the line 4x − y = 1 is perpendicular to the line 5x − k y = 2 in the coordinate plane, find the value of k.

Introduction / Context:
This question examines the relationship between slopes of perpendicular lines in coordinate geometry. When two non vertical lines are perpendicular, the product of their slopes is equal to minus one. By converting each line to slope intercept form and applying this perpendicularity condition, we can solve for the unknown parameter k that appears in the second line.

Given Data / Assumptions:
- First line: 4x − y = 1.
- Second line: 5x − k y = 2.
- The two lines are perpendicular to each other.
- We must find the value of k that makes this statement true.

Concept / Approach:
For a line written as y = mx + c, the coefficient m is the slope. For a general linear equation ax + by + c = 0, the slope is −a / b if b is non zero. Two lines with slopes m1 and m2 are perpendicular if m1 * m2 = −1. We therefore find each slope in terms of k and then use this condition to determine k.

Step-by-Step Solution:
Step 1: Rewrite the first line 4x − y = 1 in slope intercept form. Rearranging gives −y = −4x + 1, so y = 4x − 1. Step 2: The slope of the first line is therefore m1 = 4. Step 3: Rewrite the second line 5x − k y = 2. Rearranging for y gives −k y = −5x + 2, so y = (5/k)x − 2/k, provided k is not zero. Step 4: The slope of the second line is m2 = 5 / k. Step 5: Because the lines are perpendicular, their slopes satisfy m1 * m2 = −1. Substitute m1 = 4 and m2 = 5 / k to get 4 * (5 / k) = −1. Step 6: Simplify this to 20 / k = −1. Multiply both sides by k to obtain 20 = −k. Step 7: Therefore k = −20.
Verification / Alternative check:
If k = −20, then the second line is 5x + 20y = 2, or y = (−1/4)x + 1/10. The slope is −1/4. The product of slopes is 4 * (−1/4) = −1, confirming that the lines are perpendicular. Any other choice of k would give a product different from −1, so perpendicularity would not hold.

Why Other Options Are Wrong:
Option a, 20, would give a slope of 5/20 = 1/4, and the slope product 4 * 1/4 = 1, corresponding to parallel rather than perpendicular directions in terms of magnitude and orientation.
Option c and option d, 4 and −4, produce slopes of 5/4 and −5/4, neither of which satisfies m1 * m2 = −1 when multiplied by 4.
Option e, 0, would make the second line 5x = 2, a vertical line, while the first line has finite slope 4, so the slope product condition cannot be applied as written, and this is not the intended configuration.

Common Pitfalls:
Students often confuse the condition for parallel lines (equal slopes) with that for perpendicular lines (product of slopes equals minus one). Another common error is mishandling the algebra when solving for k, especially when dealing with negative signs or denominators. It is also easy to forget that k cannot be zero because that would eliminate the y term. Carefully deriving each slope and checking the product helps avoid these mistakes.

Final Answer:
The required value of k that makes the lines perpendicular is −20.