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On a city planner's coordinate map, scaled in blocks, a straight water main passes through the points A(-2, 5) and B(6, 1). A pumping station P lies on this same main directly above the location with x-coordinate 10. From P, a branch pipe runs perpendicular to the main until it reaches a reservoir located on the y-axis. What is the y-coordinate of the reservoir?
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Correct answer
B
Method 1 (line equations). First find the main's slope from A(-2, 5) and B(6, 1): slope = (1 - 5)/(6 - (-2)) = -4/8 = -1/2. The main line is y - 1 = -1/2(x - 6), which simplifies to y = -1/2 x + 4. P lies on the main at x = 10, so y = -1/2(10) + 4 = -1, giving P = (10, -1). The branch is perpendicular to the main, so its slope is the negative reciprocal of -1/2, namely +2. The branch through P(10, -1) is y - (-1) = 2(x - 10), i.e. y = 2x - 21. The reservoir sits on the y-axis, so set x = 0: y = -21.
Method 2 (vector stepping, no second equation needed). Locate P = (10, -1) as above. On the perpendicular, slope +2 means rise 2 for every run 1. To go from P to the y-axis the run is 0 - 10 = -10 (ten blocks left), so the rise is 2 × (-10) = -20. The reservoir's y-coordinate is -1 + (-20) = -21. Same answer.
Why the wrong choices tempt: choice C (-1) is the y-coordinate of P itself, the trap of answering the intermediate point rather than the reservoir. Choice D (4) comes from forgetting to turn perpendicular and reusing the main's slope, which just reproduces the main line and its intercept of 4. Choice E (19) keeps the wrong sign on the perpendicular slope (-2 rather than +2), the natural sign slip when taking the negative reciprocal. Choice A (-29) carries the main's constant term as -4 instead of +4, misplacing P at (10, -9) before an otherwise correct branch.
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