GMAT Quant Question
Set S contains points whose abscissa (x coordinate of the point) and ordinate (y coordinate of the point) 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?
The correct answer is Choice E. 3 points.
Main Idea to solve this GMAT Quant Question
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).
Step 1: Compute the equation of the line joining points (8, 0) and (0, 12)
Or the equation is 12x + 8y = 96 or 3x + 2y = 24.
We know the elements of set S contain points whose abscissa and ordinate are both natural numbers.
The equation of the line is 3x + 2y = 24 and hence, x will take even values while y will take values that are multiples of 3.
When you increment ‘x’ by k, 3x will get incremented by 3k. Unless 2y decreases by 3k, the sum will not 24.
So, 2y should be a multiple of 3. Because y is an integer, unless y is a multiple of 3, 2y will not be a multiple of 3.
The logic for why x should be an even number is also the same.
Step 2: What values will x and y take?
The values are x = 2, y = 9; x = 4, y = 6; x = 6, y = 3.
Hence, there are 3 such points that exist in set S.
The answer to this GMAT Quant sample question in coordinate geometry is option E.