Question

Let $C$ be the graph of $xy=1,$ and denote by $C^*$ the reflection of $C$ in the line $y=2x.$ Let the equation of $C^*$ be written in the form

$12x^2+bxy+cy^2+d=0.$

Find the product $bc.$

Answer

Answer: 84

Concepts:reflection (geometry)transformation hyperbola

Difficulty rating: 2170

Small Hint:

Find the reflection matrix for the line spanned by $(1,2)$

Big Hint:

Because reflection is its own inverse, substitute the reflected coordinates into $uv=1$

Solution:

Reflection in the line spanned by $(1,2)$ has matrix $\frac15\begin{pmatrix}-3&4\\4&3\end{pmatrix}.$ Thus a point $(x,y)$ on $C^*$ came from $u=\frac{-3x+4y}{5}$ and $v=\frac{4x+3y}{5}$ on $C.$ Substituting $uv=1$ gives $-12x^2+7xy+12y^2-25=0,$ or $12x^2-7xy-12y^2+25=0.$ Hence $b=-7,$ $c=-12,$ and $bc=84.$

1988 AIME Problem 14

Answer: 84

Difficulty rating: 2170

Small Hint:

Big Hint:

Solution:

Problem 14 in Other Years