Given: lines y = 3x + 1, y = -2x + 5, and a third line L1: y = mx + c with m > 0 and c a constant. L1 moves parallel to itself intersecting the two given lines in points P1 and P2 respectively. The locus of mid-points of segment P1P2 is line L2. Find the slope of L1 so that L2 has a slope that is undefined.
Here’s the scene. If we can find two points on L1, we can get its slope m. One point, clearly, is where the two given lines intersect. Call it P3. There, P1 = P2 (= P3) and the midpoint is P3 also. So: 3x + 1 = -2x + 5 → 5x = 4 → x = 4/5 and y = 17/5. P3 is (4/5, 17/5).
Since we want L2 to be vertical, another easy point to look at on L2 is the intersection of L2 with the x-axis, the point (4/5, 0), which is the midpoint between two as-yet-unknown points P1(x1, 3x1 + 1) and P2(x2, -2x2 + 5). If we can find the coordinates of P2, then we’ll have the coordinates of two points on L1 and can find the slope of the line.
Using the midpoint formula, we get two equations in x1 and x2:
(x1 + x2)/2 = 4/5
(3x1 + 1 -2x2 + 5)/2 = 0
Solving these two simultaneous equations (some algebra here), we get x1 = -14/25 and x2 = 54/25. Now we can find our points P1 on y = 3x + 1 and P2 on y = -2x + 5:
P1 = (-14/25, -17/25)
P2 = (54/25, 17/25)
Using the slope formula, we find the slope of L1 to be:
m = [17/25 – (-17/25)] / [54/25 – (-14/25)] = 34/68 = 1/2.
The answer is (b). If the slope of L1 is 1/2, then the slope of L2 is undefined.