Hi Thanks for yourvisit & interest here. Let me wait for some more days. We hv lots of stalwarts among IL folks who are passionate of maths to Derive solutions. Regards.
Thanks for your visit @svpriya . I remember your passion for maths puzzle/problems. How do you approach in solving this. Regards.
cant see the full text of the question, but towards the end, is it saying that using square roots /hypotenuse is ugly solution? huhh.. I just straight went to that a square + b square=c square But Id still like to participate - mid way of 10 (5,5) gave me the smallest size of the two pipes.
Got x = 3.636. Solution is too tedious to type out here but the main steps were to form an equation for the total length of pipes using the hypotenuse. Take the derivative, set it to 0 and solve for x.
From westvilli Stretch the normal to other side of the river in North upto 4 units. From that north end draw a straight line to meet Eastvilli on south. This line intersects at x on the river. The pump needs to be fixed at this X so that sum total length of pipes from pumping station to eastvilli and westvilli is minimum. thanks to @Gauri03 her answer is correct. Drawn to scale this illustration would produce almost same result as that of sister Gauri . 
