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Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1988, Question 14, based on Reflection.

## Reflection Problem – AIME I, 1988

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.

• is 107
• is 84
• is 840
• cannot be determined from the given information

### Key Concepts

Geometry

Equation

Algebra

But try the problem first…

Source

AIME I, 1988, Question 14

Coordinate Geometry by Loney

## Try with Hints

First hint

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

Second Hint

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

Final Step

which when expanded becomes

$12x’^{2}-7x’y’-12y’^{2}+25=0$

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