Difference between revisions of "2011 AIME I Problems/Problem 3"

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== Problem ==
 
== Problem ==
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Let <math>L</math> be the line with slope <math>\frac{5}{12}</math> that contains the point <math>A=(24,-1)</math>, and let <math>M</math> be the line perpendicular to line <math>L</math> that contains the point <math>B=(5,6)</math>.  The original coordinate axes are erased, and line <math>L</math> is made the <math>x</math>-axis and line <math>M</math> the <math>y</math>-axis.  In the new coordinate system, point <math>A</math> is on the positive <math>x</math>-axis, and point <math>B</math> is on the positive <math>y</math>-axis.  The point <math>P</math> with coordinates <math>(-14,27)</math> in the original system has coordinates <math>(\alpha,\beta)</math> in the new coordinate system.  Find <math>\alpha+\beta</math>.
 
Let <math>L</math> be the line with slope <math>\frac{5}{12}</math> that contains the point <math>A=(24,-1)</math>, and let <math>M</math> be the line perpendicular to line <math>L</math> that contains the point <math>B=(5,6)</math>.  The original coordinate axes are erased, and line <math>L</math> is made the <math>x</math>-axis and line <math>M</math> the <math>y</math>-axis.  In the new coordinate system, point <math>A</math> is on the positive <math>x</math>-axis, and point <math>B</math> is on the positive <math>y</math>-axis.  The point <math>P</math> with coordinates <math>(-14,27)</math> in the original system has coordinates <math>(\alpha,\beta)</math> in the new coordinate system.  Find <math>\alpha+\beta</math>.
  
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Thus, we have that <math>\alpha=-\frac{123}{13}</math> and that <math>\beta=\frac{526}{13}</math>. It follows that <math>\alpha+\beta=-\frac{123}{13}+\frac{526}{13}=\frac{403}{13}=\boxed{031}</math>.
 
Thus, we have that <math>\alpha=-\frac{123}{13}</math> and that <math>\beta=\frac{526}{13}</math>. It follows that <math>\alpha+\beta=-\frac{123}{13}+\frac{526}{13}=\frac{403}{13}=\boxed{031}</math>.
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== Note ==
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Since AIME only accepts nonnegative integer solutions up to <math>999</math>, once we find the distances, since the numerator of the sum of the absolute values of the abscissa and ordinate is not divisible by <math>13</math> and therefore cannot be a valid solution, the answer must be the difference instead.
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==Video Solution==
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https://www.youtube.com/watch?v=_znugFEst6E&t=919s
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~Shreyas S
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2011|n=I|num-b=2|num-a=4}}
 
{{AIME box|year=2011|n=I|num-b=2|num-a=4}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 00:58, 12 July 2022

Problem

Let $L$ be the line with slope $\frac{5}{12}$ that contains the point $A=(24,-1)$, and let $M$ be the line perpendicular to line $L$ that contains the point $B=(5,6)$. The original coordinate axes are erased, and line $L$ is made the $x$-axis and line $M$ the $y$-axis. In the new coordinate system, point $A$ is on the positive $x$-axis, and point $B$ is on the positive $y$-axis. The point $P$ with coordinates $(-14,27)$ in the original system has coordinates $(\alpha,\beta)$ in the new coordinate system. Find $\alpha+\beta$.

Solution

Given that $L$ has slope $\frac{5}{12}$ and contains the point $A=(24,-1)$, we may write the point-slope equation for $L$ as $y+1=\frac{5}{12}(x-24)$. Since $M$ is perpendicular to $L$ and contains the point $B=(5,6)$, we have that the slope of $M$ is $-\frac{12}{5}$, and consequently that the point-slope equation for $M$ is $y-6=-\frac{12}{5}(x-5)$.


Converting both equations to the form $0=Ax+By+C$, we have that $L$ has the equation $0=5x-12y-132$ and that $M$ has the equation $0=12x+5y-90$. Applying the point-to-line distance formula, $\frac{|Ax+By+C|}{\sqrt{A^2+B^2}}$, to point $P$ and lines $L$ and $M$, we find that the distance from $P$ to $L$ and $M$ are $\frac{526}{13}$ and $\frac{123}{13}$, respectively.


Since $A$ and $B$ lie on the positive axes of the shifted coordinate plane, we may show by graphing the given system that point P will lie in the second quadrant in the new coordinate system. Therefore, the abscissa of $P$ is negative, and is therefore $-\frac{123}{13}$; similarly, the ordinate of $P$ is positive and is therefore $\frac{526}{13}$.

Thus, we have that $\alpha=-\frac{123}{13}$ and that $\beta=\frac{526}{13}$. It follows that $\alpha+\beta=-\frac{123}{13}+\frac{526}{13}=\frac{403}{13}=\boxed{031}$.

Note

Since AIME only accepts nonnegative integer solutions up to $999$, once we find the distances, since the numerator of the sum of the absolute values of the abscissa and ordinate is not divisible by $13$ and therefore cannot be a valid solution, the answer must be the difference instead.

Video Solution

https://www.youtube.com/watch?v=_znugFEst6E&t=919s

~Shreyas S

See also

2011 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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