Set S contains points whose abscissa and ordinate are both natural numbers Point P an element in set......

Set S contains points whose abscissa and ordinate are both natural numbers. Point P, an element in set S has the property that the sum of the distances from point P to the point (3,0) and the point (0,5) is the lowest among all elements in set S. What is the sum of abscissa and ordinate of point P?

Any point on the line x/3 + y/5 = 1 will have the shortest overall distance.
However, we need to have integral coordinates.
So, we need to find points with integral coordinates as close as possible to the line 5x + 3y = 15.
Substitute x =1, we get y = 2 or 3
Substitute x = 2, we get y = 1 or 2
Sum of distances for (1, 2) = √8 + √10
Sum of distances for (1, 3) = √13 + √5
Sum of distances for (2, 1) = √2 + √20
Sum of distances for (2, 2) = √5 + √13
√5 + √13 is the shortest distance.
Sum of abscissa + ordinate = 4

By: Amit Kumar