The straight line passing through the points (−2, 5) and (6, b) is perpendicular to the line given by 20x + 5y = 3. Using the concept of slopes of perpendicular lines, what is the value of b?

Introduction / Context:
This coordinate geometry question tests your understanding of slopes of straight lines and the condition for perpendicularity. Two non vertical lines are perpendicular if the product of their slopes is -1. By finding the slope of the given line from its equation and relating it to the slope of the line through the two points, you can determine the unknown coordinate b.

Given Data / Assumptions:

• A line L1 passes through points (−2, 5) and (6, b).
• A line L2 is given by the equation 20x + 5y = 3.
• L1 is perpendicular to L2.
• We need to find the value of b.

Concept / Approach:
First, find the slope of the line 20x + 5y = 3 by rewriting it in slope intercept form y = mx + c. Then use the fact that if two lines are perpendicular, their slopes m1 and m2 satisfy m1 * m2 = -1. Next, compute the slope of the line passing through (−2, 5) and (6, b) in terms of b. Equating the product of slopes to -1 gives an equation in b that we can solve easily.

Step-by-Step Solution:
Rewrite the line 20x + 5y = 3 in the form y = mx + c. Solve for y: 5y = -20x + 3. Divide by 5: y = -4x + 3/5. So the slope of this line (L2) is m2 = -4. Let the slope of line L1, passing through (−2, 5) and (6, b), be m1. Slope m1 = (b - 5) / (6 - (−2)) = (b - 5) / 8. Since L1 is perpendicular to L2, m1 * m2 = -1. So (b - 5) / 8 * (−4) = -1. Simplify: (b - 5) * (−4) / 8 = -1. Note that −4 / 8 = −1/2, so (b - 5) * (−1/2) = -1. Multiply both sides by -2: b - 5 = 2. Therefore, b = 5 + 2 = 7.

Verification / Alternative check:
Compute slopes explicitly with b = 7. The slope of L1 is (7 - 5) / (6 - (−2)) = 2 / 8 = 1/4. The slope of L2 is -4. Their product is (1/4) * (−4) = -1, which confirms that L1 and L2 are perpendicular when b = 7.

Why Other Options Are Wrong:
If b were -7, 4, -4, or 0, the slope (b - 5)/8 would not be 1/4, and the product with -4 would not equal -1. For example, if b = 4, the slope would be (4 - 5)/8 = -1/8, and the product with -4 would be 1/2, not -1. Hence those values do not satisfy the perpendicularity condition.

Common Pitfalls:
Typical errors include miscomputing the slope from the standard form equation, forgetting that the product of slopes for perpendicular lines is -1 (not 1), or mixing up the coordinates in the slope formula. Carefully rewriting the given line in y = mx + c form and plugging points correctly into the slope formula avoids these issues.

Final Answer:
The value of b that makes the line through (−2, 5) and (6, b) perpendicular to 20x + 5y = 3 is 7.