Question:

Under a certain transformation the image of the point (x,y) is (X,Y) where $\displaystyle \begin{pmatrix} X\\ Y \end{pmatrix} = \begin{pmatrix} 1 & 4\\ 2 & 3 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} $. This transformation maps any point on the line $\displaystyle y = mx $ onto another point on the line $\displaystyle y=mx$. Find the two possible values of m.

My attempt:

I tried to look at the the effect of the transformation on the unit square $\displaystyle \begin{pmatrix} X\\ Y \end{pmatrix} = \begin{pmatrix} 1 & 4\\ 2 & 3 \end{pmatrix} \begin{pmatrix} 0 &1 &1 &0 \\ 0 &0 & 1 & 1 \end{pmatrix}$

$\displaystyle = \begin{pmatrix} 0 &1 &5 &4 \\ 0 &2 & 5 & 3 \end{pmatrix}$

Since it says y= mx

i thought it had something to do with reflection in the line y=mx where $\displaystyle m = tan \alpha $

I have tried to calculate the angle $\displaystyle \alpha $ between i.e ( 1,0) and its image (1,2) or (0,1) (4,3) but it could not figure it out.

I would like to know if my approach is wrong or right?

Please can i have help on finding the two values of m