Reflection Problem | AIME I, 1988 | Question 14

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.

Let P(x,y) on C such that P'(x’,y’) on C’ where both points lie on the line perpendicular to y=2x

slope of PP’=\(\frac{-1}{2}\), then \(\frac{y’-y}{x’-x}\)=\(\frac{-1}{2}\)

or, x’+2y’=x+2y

also midpoint of PP’, \((\frac{x+x’}{2},\frac{y+y’}{2})\) lies on y=2x

or, \(\frac{y+y’}{2}=x+x’\)

or, 2x’-y’=y-2x

solving these two equations, x=\(\frac{-3x’+4y’}{5}\) and \(y=\frac{4x’+3y’}{5}\)

putting these points into the equation C \(\frac{(-3x’+4y’)(4x’+3y’)}{25}\)=1

which when expanded becomes

\(12x’^{2}-7x’y’-12y’^{2}+25=0\)

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

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