Simple linear regression Consider the simple regression model $y=\beta_0+\beta_1x+u$, where $\text{corr}(x,u)=1$ and all random variables have normal distributions. Is it possible to provide asymptotically consistent estimates for $\beta_1$?
Since $corr(x,u)=1$, we can write  $u=ax+b$ and substitute it to get $y=\beta_0+b+(\beta_1+a)x$. So our observations lie on the straight line given by this equation. What sort of conclusion can we derive from this?
 A: This is a weird question: the simple regression model is defined conditional to the regressor $X$, hence (a) for each observation $y_1$, $x_i$ is observed as well and (b) if the noise $U$ is perfectly correlated with $X$, this means that the noise $u_i$ is also observed: $u_i=ax_i+b$. 
Conclusion: there is no randomness and one only needs two observations to derive w/o error the parameters $(\beta_0+b,\beta_1+a)$, therefore $(\beta_0,\beta_1)$ if $(a,b)$ is known.
Note: If $(a,b)$ is unknown, w/o any additional assumption, all one can estimate is $(\beta_0+b,\beta_1+a)$.
A: This is not simple linear regression. Simple linear regression is $y = \beta_0 + \beta_1x + u$, but with cor(x, u) = 0. Indeed, your second equation $y = \beta_0 + b + (\beta_1 + a)x$ is the simple linear regression for the situation, but you should add $u$ with $u \sim \mathcal{N}(0,0)$
One can conclude that this is fantasy land, because all the observations never lie exactly on the line. George Box said "all models are wrong but some models are useful". You've posited a model which is exactly right. No model error, no measurement error.
