Apart from usual and discrete metric is there a metric on R which satisfy: d(x, y) = d(x+r , y+r) where x and y are any real no. and r is arbitary real no.
Similarly is there a metric on R which satisfy: d(x, y) = d(x.r , y.r) for any real no. r except than zero.
1. Metrics that satisfy $d(x+r,y+r)=d(x,y)$: There are many metrics of this kind, as remarked by other users. Also the sum of any such metrics still has the property.
2. Metrics that satisfy $d(xr,yr)=d(x,y)$: There is such a metric on $(0,\infty)$, namely $d(x,y)=|\log(x)-\log(y)|$.
There are also such metrics on $\mathbb R$, and something can be said about them. The value of $d(0,x)$ is independent of $x$; let us call this constant $c$. The values of the metric are bounded by $2c$ due to the triangle inequality. Because of the scaling invariance there is a function $\phi:\mathbb R\setminus\{0,1\}\to(0,2c)$ such that $d(a,b)=\phi(a/b)$ when $a\neq b$. Starting from the triangle inequality, one can show various inequalities for $\phi$, but there are possibilities left.
For example, $\phi\equiv1$ (discrete metric) or $\phi(x)=1$ for $x<0$ and $\phi(x)=\min\{1,|\log(x)|\}$ when $x>0$. The second one is not discrete.