How do I get from the eigenvectors of the covariance matrix to the regression parameters?

Question

I have a linear regression problem

$$ y = a x + b$$

with errors on $x$ and $y$ that are uncorrelated and unitary and I have to find $a$ and $b$. To do this, I want to use principal component analysis (PCA):

1. Calculate the $n \times 2$ matrix $D = ( x' y' )$, where $x' = x - \langle x \rangle$ and $y' = y - \langle y\rangle$.

2. Calculate the covariance matrix of the transformed data $C = D^T D$.

3. Calculate the eigenvectors and eigenvalues of $C$ using SVD.

How do I find $a$ and $b$ from the eigenvectors of $C$?