This question is a GMAT Quant problem solving question in Coordinate Geometry. Concept Covered: basic understanding of equation of straight lines and an understanding of the coordinates of points that lie on the line. Also includes basic number property concepts to determine the coordinates of the points on the line. A GMAT hard math question - GMAT 700 level quant question.

Question 2: 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 (8, 0) and the point (0, 12) is the lowest among all elements in set S. How many such points P exist in set S?

- 1
- 5
- 11
- 8
- 3

From INR

The sum of the distances from point P to the other two points will be at its lowest only when point P lies on the line segment joining the points (8, 0) and (0, 12).

(8, 0) and (0, 12) are the coordinates of the x and y intercepts of the line respectively.

So, the equation of the line segment joining the points (8, 0) and (0, 12) is \\frac{x}{8}) + \\frac{y}{12}) = 1

Or the equation of the line is 12x + 8y = 96 or 3x + 2y = 24.

The question states that the elements of set S contain points whose abscissa and ordinate are both natural numbers. i.e., their x and y coordinates are both positive integers.

The equation of the line is 3x + 2y = 24. Positive integer values that satisfy the equation will be such that their 'x' values will be even and their 'y' values will be multiples of 3.

The values are

1. x = 2, y = 9

2. x = 4, y = 6

3. x = 6, y = 3.

Hence, there are 3 such points that exist in set S.

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