1988 AIME Problems/Problem 14

Revision as of 02:16, 17 November 2007 by Bowei (talk | contribs) (Solution)

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 $bc$.

Solution

$bc=84$

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

See also

1988 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
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