Question

# 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

CAT 2021

A
B
C
D
##### 4
Solution

Correct option is

(D)

any point on the line x/3 + y/5 = 1 will have the shortest distance in total

however , we need to have integral coordinates .

so , we need to find the points with inegral coordinates as close as possible to the line 5x + 3y = 15

put x = 1 , we get y = 2 or 3

put x = 2 , we get y = 1 or 2

sum of distance for (1,2) = $\sqrt{8} + {\sqrt{10}}$

sum of distance for (1,2) =$\sqrt{13} + {\sqrt{5}}$

sum of distance for (1,2) =$\sqrt{2} + {\sqrt{20}}$

sum of distance for (1,2) = $\sqrt{5} + {\sqrt{13}}$

$\sqrt{5} + {\sqrt{13}}$ is the shortest distance

sum of abscissa + ordinate = 4

so , D is the correct option