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

(→‎Solution 2)
(→‎Solution 2)
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Now to find the equation of the hyperbola, we multiply the two expressions together to get one side of the equation: <math>(3x-4y)(4x+3y)=12x^2-7xy-12y^2</math>. At this point, the right hand side of the equation will be determined by plugging the point <math>(\frac{\sqrt{2}}{2},\sqrt{2})</math>, which is unchanged by the reflection, into the expression. But this is not necessary. We see that <math>b=-7</math>, <math>c=-12</math>, so <math>bc=\boxed{084}</math>.
 
Now to find the equation of the hyperbola, we multiply the two expressions together to get one side of the equation: <math>(3x-4y)(4x+3y)=12x^2-7xy-12y^2</math>. At this point, the right hand side of the equation will be determined by plugging the point <math>(\frac{\sqrt{2}}{2},\sqrt{2})</math>, which is unchanged by the reflection, into the expression. But this is not necessary. We see that <math>b=-7</math>, <math>c=-12</math>, so <math>bc=\boxed{084}</math>.
 +
 +
 +
== Solution 3 ==
 +
The matrix for a reflection about the polar line <math>\theta = \alpha, \alpha+\pi</math> 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 <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>.
 +
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{5}{5}y)\right</math>.
 +
 +
The new coordinates <math>(x*, y*)</math> must satisfy <math>x*y* = 1</math>. Therefore,
 +
<cmath>\left(-\frac{3}{5}x+\frac{4}{5}y)\right)\left(\frac{4}{5}x+\frac{5}{5}y)\right) = 1</cmath>
 +
<cmath>-\frac{12}{25}x^2 + \frac{7}{25}xy + \frac{12}{25}y^2 - 1 = 0</cmath>
 +
<cmath>12x^2 - 7xy - 12y^2 + 25 = 0</cmath>
 +
Thus, <math>b = -7</math> and <math>c = -12</math>, so <math>bc = 84</math>. The answer is <math>084</math>.
  
 
== See also ==
 
== See also ==

Revision as of 22:03, 25 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 $\alpha = \arctan 2$. 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 $y* = \frac{4}{5}x+\frac{5}{5}y)\right$ (Error compiling LaTeX. Unknown error_msg).

The new 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)
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Problem 13 		Followed by
Problem 15
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