Problem:
The graph of 2x2+xy+3y2−11x−20y+40=0 is an ellipse in the first quadrant of the xy-plane. Let a and b be the maximum and minimum values of xy over all points (x,y) on the ellipse. What is the value of a+b ?
Answer Choices:
A. 3
B. 10
C. 27
D. 29
E. 214 Solution:
A line y=mx intersects the ellipse in 0,1 , or 2 points. The intersection consists of exactly one point if and only if m=a or m=b. Thus a and b are the values of m for which the system
2x2+xy+3y2−11x−20y+40=0y=mx
has exactly one solution. Substituting mx for y in the first equation gives
2x2+mx2+3m2x2−11x−20mx+40=0
or, by rearranging the terms,
(3m2+m+2)x2−(20m+11)x+40=0
The discriminant of this equation is
(20m+11)2−4⋅40⋅(3m2+m+2)=−80m2+280m−199
which must be zero if m=a or m=b. Thus a+b is the sum of the roots of the equation −80m2+280m−199=0, which is 80280=27.