Convert linear-fit results onto an orthogonal basis

I have the results of the linear fit of some experimental data:

 y=a0+a1.x

together with the covariance matrix: sigma_square_a0, sigma_square_a1, sigma_a0_a1. I d like to express the results using an orthogonal polynomial basis, that is to say as

 y=b0+b1.(x-c)

with sigma_b_c=0 and sigma_square_c=0. I bet it is just a matter or diagonalizing the covariance matrix but i don't see how to define the c.

Any thoughts?