$T $ is a Reflection about $y=mx$ axis along $y=\frac{-x}{m}$ then $T(a,b)=\Big(2(\frac{a+mb}{1+m^{2}})-a,2(\frac{m(a+mb)}{1+m^{2}})-b\Big)\forall (a,b)\in \mathbb{R^{2}}$. How can I derive the Result?
I don't Know How to start analytically. Please help me to find the formula.
Let the image be $(a^*, b^*)$. This point as well as the original point $(a, b)$ lie on the a line $L$ with slope $\frac{-1}{m}$ and some intercept (call it $c$). Solving for $c$, $c=b+\frac{a}{m}$.
Where does $L$ intersect the original line $y=mx$? At the point
$$(x^*,y^*)=\left(\frac{a+mb}{1+m^2} , m\frac{a+mb}{1+m^2}\right)$$
So, now, you know that the image point must be at $\left(a + 2(x^*-a), b+2(y^*-b)\right)$ which is the point you have.