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?
A. 1
B. 5
C. 11
D. 8
E. 3
The correct answer is Choice E. 3 points.
Explanatory Answer
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).
The equation of the line segment joining the points (8, 0) and (0, 12) is
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.
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.