2021 AMC 10B Fall Problem 17

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

Two reflections through intersecting lines equal a rotation

Compare the angle change from

(-1,4)

to

(4,1)

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 .

Problem 17 in other years