Question

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 $m?$

$5x+2y=0$

$3x+2y=0$

$x-3y=0$

$2x-3y=0$

$5x-3y=0$

Answer: D

Concepts:transformation slope trigonometric identity

Difficulty rating: 2150

Solution:

Two reflections across lines through the origin are equivalent to a rotation by twice the angle from the first reflecting line to the second.

The point $(-1,4)$ is sent to $(4,1),$ which is a $90^\circ$ clockwise rotation. Therefore line $m$ is $45^\circ$ clockwise from line $\ell.$

The slope of $\ell$ is $5.$ If $\theta_m=\theta_\ell-45^\circ,$ then $\tan\theta_m=\frac{5-1}{1+5}=\frac23.$

Thus line $m$ is $y=\frac23x,$ or $2x-3y=0.$

Thus, the answer is D .

2021 AMC 10B Fall Problem 17

Answer: D

Difficulty rating: 2150

Solution:

Problem 17 in Other Years