Difference between revisions of "1988 AIME Problems/Problem 14"
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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 23:03, 25 November 2015
Problem
Let be the graph of
, and denote by
the reflection of
in the line
. Let the equation of
be written in the form
Find the product .
Solution 1
Given a point on
, we look to find a formula for
on
. Both points lie on a line that is perpendicular to
, so the slope of
is
. Thus
. Also, the midpoint of
,
, lies on the line
. Therefore
.
Solving these two equations, we find and
. Substituting these points into the equation of
, we get
, which when expanded becomes
.
Thus, .
Solution 2
The asymptotes of are given by
and
. Now if we represent the line
by the complex number
, then we find the direction of the reflection of the asymptote
by multiplying this by
, getting
. Therefore, the asymptotes of
are given by
and
.
Now to find the equation of the hyperbola, we multiply the two expressions together to get one side of the equation: . At this point, the right hand side of the equation will be determined by plugging the point
, which is unchanged by the reflection, into the expression. But this is not necessary. We see that
,
, so
.
Solution 3
The matrix for a reflection about the polar line 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 . Then
, so
and
.
Therefore, if
is mapped to
under the reflection, then
and $y* = \frac{4}{5}x+\frac{5}{5}y)\right$ (Error compiling LaTeX. Unknown error_msg).
The new coordinates must satisfy
. Therefore,
Thus,
and
, so
. The answer is
.
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.