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
Check the Answer
Answer: is 84.
AIME I, 1988, Question 14
Coordinate Geometry by Loney
Try with Hints
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.
Other useful links
- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA