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

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

Geometry

Equation

Algebra

But try the problem first...

Answer: is 84.

Source

Suggested Reading

AIME I, 1988, Question 14

Coordinate Geometry by Loney

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.

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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