The value of y that minimizes the sum of the two distances from (3,5) to (1,y) and from (1,y) to (4,9) can be written as a/b where a and b are coprime positive integers. Find a+b.

Dear Vivek,

the minimum distance of the points from (1,y) is the distance from the intersection of their perpendicular bisectors to the line x=1
hence slope of perpendicular bisector=> -4=2y-14 / 2x -7
                                                           => 8x + 2y = 42.
putting x=1,y=17,
hence a+b= 17 +1 =18 (ANS).

Regards.

the minimum distance of the points from (1,y) is the distance from the intersection of their perpendicular bisectors to the line x=1
hence slope of perpendicular bisector=> -4=2y-14 / 2x -7
                                                           => 8x + 2y = 42.
putting x=1,y=17,
hence a+b= 17 +1 =18 (ANS).