Borehole Siting with Locus and Distance Constraints
A district water committee is siting a new borehole. On a map with coordinates in kilometres, village A is at (2, 1), village B at (10, 5), and a school S at (6, 7). The borehole must be exactly equidistant from A and B, and no more than 5 km from the school. A technical officer draws the perpendicular bisector of AB but does not check whether it actually crosses the 5 km circle. A second constraint is added later: a third village C at (4, 8) insists that the borehole be within 4 km of C, otherwise its representatives will block the choice. The committee wants a defensible recommendation, not just an equation. The map is flat and distances are straight-line; roads are ignored at this stage. A councillor points out that two possible points may satisfy the first two conditions and asks which one should be chosen.
1. Find the equation of the locus of points equidistant from A(2,1) and B(10,5). Explain why this locus is the perpendicular bisector of AB.
Read the worked answer
Midpoint M=(6,3). Slope AB=(5-1)/(10-2)=4/8=1/2. The perpendicular slope is -2, so the line through M is y-3=-2(x-6), i.e. y=-2x+15. Algebraically, if P(x,y) is equidistant, then (x-2)^2+(y-1)^2=(x-10)^2+(y-5)^2. Expanding gives x^2+y^2-4x-2y+5=x^2+y^2-20x-10y+125, so 16x+8y=120 and 2x+y=15. This is exactly the perpendicular bisector, because equal distances to the endpoints of segment AB characterise that line.
2. Show that this locus intersects the circle of radius 5 km centred on S(6,7). Find the coordinates of the two possible borehole sites, correct to 2 decimal places.
Read the worked answer
Circle: (x-6)^2+(y-7)^2=25. Substitute y=-2x+15, so y-7=-2x+8=-2(x-4). Then (x-6)^2+4(x-4)^2=25, which simplifies to x^2-12x+36+4x^2-32x+64=25, i.e. 5x^2-44x+75=0. The discriminant is 44^2-4(5)(75)=1936-1500=436>0, so the line does meet the circle. x=(44±sqrt436)/10≈(44±20.881)/10. Thus x≈6.49 or x≈2.31. The corresponding y-values are y=-2x+15, giving y≈2.02 and y≈10.38. The sites are P≈(6.49,2.02) and Q≈(2.31,10.38).
3. Village C is at (4,8) and requires the borehole to be within 4 km of C. Test both sites and recommend one, with reasoning. Does the recommendation satisfy all stated constraints?
Read the worked answer
For P≈(6.49,2.02): CP^2≈(2.49)^2+(-5.98)^2≈6.20+35.76=41.96, so CP≈6.48 km, which is more than 4 km. For Q≈(2.31,10.38): CQ^2≈(-1.69)^2+(2.38)^2≈2.86+5.66=8.52, so CQ≈2.92 km, which is within 4 km. Recommend Q. It lies on the perpendicular bisector, so it is equidistant from A and B; it lies on the 5 km circle around S by construction; and it is within 4 km of C. P fails the C constraint. The recommendation satisfies all three stated conditions, with rounding accepted.