Yummy Straight Lines

The line $y=x$ forms angle $\theta_1=\frac{\pi}{4}$ with the positive $x$ axis. Let $\theta_2$ denote the angle between $y=x$ and $y=2x$ . Denote the line we are to find by $L_3:\, y=mx$ and let $\theta_3$ denote the angle between $L_3$ and the positive $x$ axis.

We have $\theta_2=\arctan{2}-\frac{\pi}{4}$ and thus $\theta_3=2\arctan{2}-\frac{\pi}{4}$ .

Therefore $m=\tan{\theta_3}=\frac{\tan(2\arctan{2})-1}{1+\tan(2\arctan{2})}$

$\Rightarrow\quad m=\frac{ \frac{4}{1-4}-1 }{ 1+\frac{4}{1-4} } = 7.$

Alternatively, we can approach the whole problem from a vectors points of view:

Consider the reflection of the point with position vector $p=\pmatrix{1\\1}$ in the line $r=\lambda\pmatrix{1\\2}$ . Denote by $q$ the position vector of the point on the line closest to $p$ . We have $\pmatrix{1\\2}\cdot\pmatrix{\lambda-1 \\ 2\lambda-1}=0$ from which we deduce $\lambda=0.6$ . That is, $q=\pmatrix{0.6 \\ 1.2}.$

Thus the image of $p$ after reflection is $\pmatrix{1 \\ 1}+2\pmatrix{-0.4 \\ 0.2}=\pmatrix{0.2\\1.4} .$

Therefore, since the origin is invariant under this transformation, the image line has equation $y=7x$ .

The line y = x makes an inclination of 45 degrees with x axis. The line y = 2x makes an inclination of Arctan(2) with x axis which is equal to ~ 63.435 degrees.
The angle between y=2x and y=x is 18.435 degrees.

If the line y = 2x is angular bisector of y= x and y = mx ,then y=mx should make an inclination of (63.435 + 18.435) degrees with x-axis.

Therefore m should be equal to tan(63.435 + 18.435) or 7

Therefore y= 2x is angular bisector of y=x and y=7x.