I tried doing this by coordinate geometry, taking the origin as the midpoint between the houses. Then the houses are at the points (±150,0) and the well is at (50,400). If the junction is at (x,y) then the sum of the distances from (x,y) to the houses is 400. So (x,y) lies on an ellipse with foci at the houses, major axis 400 and eccentricity 3/4. The equation of the ellipse is , and we want to find the shortest distance from (50,400) to a point on the ellipse.
That's where I get stuck. The shortest distance seems to be given by a quartic equation with horrendous coefficients, and I can see no way of solving it. A bit of numerical trial and error seems to show that the minimum distance is about 269.9 yards, when the junction is somewhere near the point (26,131).
Is there some trick here that I'm missing, or have I made some silly mistake?