Difference between revisions of "2021 Fall AMC 10B Problems/Problem 17"

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==Solution==
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==Problem==
 
Distinct lines <math>\ell</math> and <math>m</math> lie in the <math>xy</math>-plane. They intersect at the origin. Point <math>P(-1, 4)</math> is reflected about line <math>\ell</math> to point <math>P'</math>, and then <math>P'</math> is reflected about line <math>m</math> to point <math>P''</math>. The equation of line <math>\ell</math> is <math>5x - y = 0</math>, and the coordinates of <math>P''</math> are <math>(4,1)</math>. What is the equation of line <math>m?</math>
 
Distinct lines <math>\ell</math> and <math>m</math> lie in the <math>xy</math>-plane. They intersect at the origin. Point <math>P(-1, 4)</math> is reflected about line <math>\ell</math> to point <math>P'</math>, and then <math>P'</math> is reflected about line <math>m</math> to point <math>P''</math>. The equation of line <math>\ell</math> is <math>5x - y = 0</math>, and the coordinates of <math>P''</math> are <math>(4,1)</math>. What is the equation of line <math>m?</math>
  
 
<math>(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0\qquad(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0</math>
 
<math>(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0\qquad(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0</math>
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==See Also==
 
==See Also==
 
{{AMC10 box|year=2021 Fall|ab=B|num-a=18|num-b=16}}
 
{{AMC10 box|year=2021 Fall|ab=B|num-a=18|num-b=16}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 12:02, 23 November 2021

Problem

Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. What is the equation of line $m?$

$(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0\qquad(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0$

See Also

2021 Fall AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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All AMC 10 Problems and Solutions

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