1988 AIME Problems/Problem 14

Problem

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 $bc$ .

$bc=84$

We need to find a formula for $P^*$ if $P=(x,y)$ .

With some algebra, we find this to be:

$P^*=(\frac{-3x+4y}{5},\frac{4x+3y}{5})$ .

Substituting in, we get:

$\frac{(-3x+4y)(4x+3y)}{25}=1$

Expanding,

$12x^2-7xy-12y^2+25=0$ .

Thus, $bc=(-7)(-12)=84.$

1988 AIME (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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