Problem:
A silo (right circular cylinder) with diameter 20 meters stands in a field. MacDonald is located 20 meters west and 15 meters south of the center of the silo. McGregor is located 20 meters east and g>0 meters south of the center of the silo. The line of sight between MacDonald and McGregor is tangent to the silo. The value of g can be written as dab−c, where a,b,c,d are positive integers, b is not divisible by the square of any prime, and d is relatively prime to gcd(a,c). What is a+b+c+d?
Answer Choices:
A. 119
B. 120
C. 121
D. 122
E. 123
Solution:
Let the silo's center be at the origin O(0,0), and its radius be R=10.
MacDonald is located at M(−20,−15), and McGregor is located at N(20,−g) with g>0.
The slope of their line of sight is
m=20−(−20)−g−(−15)=4015−g.
Equation of the line through (−20,−15) is y+15=m(x+20), which simplifies to −mx+y+(15−20m)=0.
Since the line is tangent to the circle x2+y2=100, the perpendicular distance from the origin to the line equals the radius: