Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1988 AIME Problems/Problem 14"

(→‎Solution 3)
(→‎Solution 3)
Line 27: Line 27:
 
This is not hard to derive using a basic knowledge of linear transformations. You can refer here for more information: https://en.wikipedia.org/wiki/Orthogonal_matrix
 
This is not hard to derive using a basic knowledge of linear transformations. You can refer here for more information: https://en.wikipedia.org/wiki/Orthogonal_matrix
  
−
Let <math>\alpha = \arctan 2</math>. Then <math>2\alpha = \arctan\left(-\frac{4}{3}\right)</math>, so <math>\cos(2\alpha) = -\frac{3}{5}</math> and <math>\sin(2\alpha) = \frac{4}{5}</math>.
+
Let <math>\alpha = \arctan 2</math>. Note that the line of reflection, <math>y = 2x</math>, is the polar line <math>\theta = \alpha, \alpha+\pi</math>. Then <math>2\alpha = \arctan\left(-\frac{4}{3}\right)</math>, so <math>\cos(2\alpha) = -\frac{3}{5}</math> and <math>\sin(2\alpha) = \frac{4}{5}</math>.
 +
 
 
Therefore, if <math>(x', y')</math> is mapped to <math>(x, y)</math> under the reflection, then <math>x = -\frac{3}{5}x'+\frac{4}{5}y'</math> and <math>y = \frac{4}{5}x'+\frac{3}{5}y'</math>. Since the transformation matrix represents a reflection, it must be its own inverse; therefore, <math>x' = -\frac{3}{5}x+\frac{4}{5}y</math> and <math>y' = \frac{4}{5}x+\frac{3}{5}y</math>.
 
Therefore, if <math>(x', y')</math> is mapped to <math>(x, y)</math> under the reflection, then <math>x = -\frac{3}{5}x'+\frac{4}{5}y'</math> and <math>y = \frac{4}{5}x'+\frac{3}{5}y'</math>. Since the transformation matrix represents a reflection, it must be its own inverse; therefore, <math>x' = -\frac{3}{5}x+\frac{4}{5}y</math> and <math>y' = \frac{4}{5}x+\frac{3}{5}y</math>.
  

Revision as of 02:43, 26 November 2015

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 1

Given a point $P (x,y)$ on $C$, we look to find a formula for $P' (x', y')$ on $C^*$. Both points lie on a line that is perpendicular to $y=2x$, so the slope of $\overline{PP'}$ is $\frac{-1}{2}$. Thus $\frac{y' - y}{x' - x} = \frac{-1}{2} \Longrightarrow x' + 2y' = x + 2y$. Also, the midpoint of $\overline{PP'}$, $\left(\frac{x + x'}{2}, \frac{y + y'}{2}\right)$, lies on the line $y = 2x$. Therefore $\frac{y + y'}{2} = x + x' \Longrightarrow 2x' - y' = y - 2x$.

Solving these two equations, we find $x' = \frac{-3x + 4y}{5}$ and $y' = \frac{4x + 3y}{5}$. Substituting these points into the equation of $C$, we get $\frac{(-3x+4y)(4x+3y)}{25}=1$, which when expanded becomes $12x^2-7xy-12y^2+25=0$.

Thus, $bc=(-7)(-12)=\boxed{084}$.

Solution 2

The asymptotes of $C$ are given by $x=0$ and $y=0$. Now if we represent the line $y=2x$ by the complex number $1+2i$, then we find the direction of the reflection of the asymptote $x=0$ by multiplying this by $2-i$, getting $4+3i$. Therefore, the asymptotes of $C^*$ are given by $4y-3x=0$ and $3y+4x=0$.

Now to find the equation of the hyperbola, we multiply the two expressions together to get one side of the equation: $(3x-4y)(4x+3y)=12x^2-7xy-12y^2$. At this point, the right hand side of the equation will be determined by plugging the point $(\frac{\sqrt{2}}{2},\sqrt{2})$, which is unchanged by the reflection, into the expression. But this is not necessary. We see that $b=-7$, $c=-12$, so $bc=\boxed{084}$.


Solution 3

The matrix for a reflection about the polar line $\theta = \alpha, \alpha+\pi$ is: \[ \left[ \begin{array}{ccc} \cos(2\alpha) & \sin(2\alpha) \\ \sin(2\alpha) & -\cos(2\alpha) \end{array} \right] \] This is not hard to derive using a basic knowledge of linear transformations. You can refer here for more information: https://en.wikipedia.org/wiki/Orthogonal_matrix

Let $\alpha = \arctan 2$. Note that the line of reflection, $y = 2x$, is the polar line $\theta = \alpha, \alpha+\pi$. Then $2\alpha = \arctan\left(-\frac{4}{3}\right)$, so $\cos(2\alpha) = -\frac{3}{5}$ and $\sin(2\alpha) = \frac{4}{5}$.

Therefore, if $(x', y')$ is mapped to $(x, y)$ under the reflection, then $x = -\frac{3}{5}x'+\frac{4}{5}y'$ and $y = \frac{4}{5}x'+\frac{3}{5}y'$. Since the transformation matrix represents a reflection, it must be its own inverse; therefore, $x' = -\frac{3}{5}x+\frac{4}{5}y$ and $y' = \frac{4}{5}x+\frac{3}{5}y$.

The original coordinates $(x', y')$ must satisfy $x'y' = 1$. Therefore, \[\left(-\frac{3}{5}x+\frac{4}{5}y\right)\left(\frac{4}{5}x+\frac{5}{5}y\right) = 1\] \[-\frac{12}{25}x^2 + \frac{7}{25}xy + \frac{12}{25}y^2 - 1 = 0\] \[12x^2 - 7xy - 12y^2 + 25 = 0\] Thus, $b = -7$ and $c = -12$, so $bc = 84$. The answer is $084$.

See also

1988 AIME (Problems • Answer Key • Resources)
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1988_AIME_Problems/Problem_14&oldid=73198"