Problem: The coordinates of A,B and C are (5,5),(2,1) and (0,k) respectively. The value of k that makes AC+BC as small as possible is:
Answer Choices:
A. 3
B. 421
C. 376
D. 465
E. 271
Solution:
The smallest possible value of AC+BC is obtained when C is the intersection of the y-axis, with the line that leads from A to the mirror image (the mirror being the y-axis) B′:(−2,1) of B. This is true because CB′=CB and a straight line is the shortest path between two points. The line through A and B′ is given by
y=5+25−1x+k=74x+k
To find k, we use the fact that the line goes through A :
5=74⋅5+k,k=5−720=715=271:
∴C has coordinates (0,271).