2004 AMC 12B Problem 21

Problem 21 of 25HarderAlgebra Geometry

The graph of

2x^2 + xy + 3y^2

- 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

\dfrac{y}{x}

over all points

(x, y)

on the ellipse. What is the value of

a + b?

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Solution

The slopes

a

and

b

are the values of

m

for which

y = mx

meets the ellipse in exactly one point. Substituting gives

\begin{aligned} &(3m^2 + m + 2)x^2 \\ &\quad {}- (20m + 11)x + 40 = 0. \end{aligned}

Setting its discriminant to zero yields

-80m^2 + 280m - 199 = 0.

By Vieta’s formulas,

a + b = \dfrac{280}{80} = \dfrac{7}{2}.

Thus, the correct answer is C.