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 $m?$

$5x+2y=0$

$3x+2y=0$

$x-3y=0$

$2x-3y=0$

$5x-3y=0$

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

Solution:

Problem 17 in Other Years