To clarify notation, you have an universe of $n=2000 \space$ stocks and two portfolio vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{n}$ with $\left\|\mathbf{a}\right\|_{1}=\left\|\mathbf{b}\right\|_{1}=1$. Further, you have Estimators for the true Variance $\operatorname{Var}\left[\mathbf{a}\right]$ resp. $\operatorname{Var}\left[\mathbf{b}\right]$ and the Covariance $\operatorname{Cov}\left[\mathbf{a},\mathbf{b}\right]$.
Normally you'd multiply the portfolio weights with the random return vector $\boldsymbol{\mu}\in\mathbb{R}^{n}$ to get the portfolio returns $p_{\mathbf{a}} := \mathbf{a}^{T}\boldsymbol{\mu}$. This portfolio return now is a random variable and it makes sense to speak of its Variance etc. In slight abuse of Notation you can call the portfolio value $a$, which is now a real number. The same holds for the portfolio value $b$.
For a new portfolio $\alpha{a}+\beta{b}=:{c}\in\mathbb{R}\space$ with weights $\alpha+\beta=1$ you want to know $\operatorname{Var}\left[{c}\right]$.
For two correlated random vectors the following holds (http://en.wikipedia.org/wiki/Covariance#Properties):
$$\operatorname{Var}\left[{c}\right] = \operatorname{Var}\left[\alpha{a}+\beta{b}\right] = \alpha^{2}\operatorname{Var}\left[{a}\right] + \beta^{2}\operatorname{Var}\left[{b}\right] + 2\alpha\beta\operatorname{Cov}\left[{a},{b}\right]$$
You can also see this if you write down the quadratic form of your Variance-Covariance-Matrix of portfolios ${a}, {b}$ with the corresponding weighting vector that forms portfolio ${c}$. For that, note that
$$
{c} = \begin{pmatrix}
\alpha\\\beta
\end{pmatrix}^{T}
\begin{pmatrix}
{a}
\\
{b}
\end{pmatrix}
= \alpha{a}+\beta{b}
$$
holds. Therefore you can write
$$\operatorname{Var}\left[{c}\right]
=
\operatorname{Var}\left[
\begin{pmatrix}
\alpha\\\beta
\end{pmatrix}^{T}
\begin{pmatrix}
{a}
\\
{b}
\end{pmatrix}\right]
\overset{(1)}{=}
\begin{pmatrix}
\alpha\\\beta
\end{pmatrix}^{T}\operatorname{Var}\left[
\begin{pmatrix}
{a}
\\
{b}
\end{pmatrix}\right]
\begin{pmatrix}
\alpha\\\beta
\end{pmatrix}$$ $$
=
\begin{pmatrix}
\alpha\\\beta
\end{pmatrix}^{T}
\begin{pmatrix}
\operatorname{Var}\left[{a}\right] & \operatorname{Cov}\left[{a},{b}\right]
\\
\operatorname{Cov}\left[{a},{b}\right] & \operatorname{Var}\left[{b}\right]
\end{pmatrix}
\begin{pmatrix}
\alpha\\\beta
\end{pmatrix}$$ $$
=
\alpha^{2}\operatorname{Var}\left[{a}\right] + \beta^{2}\operatorname{Var}\left[{b}\right] + 2\alpha\beta\operatorname{Cov}\left[{a},{b}\right]\textrm{,}
$$
with (1) the matrix-wise principle on how to get a constant out of the Variance, see (http://en.wikipedia.org/wiki/Covariance#A_more_general_identity_for_covariance_matrices).
You are interested in
$$
\operatorname{Cov}\left[{a},{c}\right]
= \operatorname{Cov}\left[{a},\alpha{a}+\beta{b}\right]$$ $$
\overset{(2)}{=} \alpha\operatorname{Cov}\left[a,a\right]+\beta\operatorname{Cov}\left[a,b\right]
=
\alpha\operatorname{Var}\left[a\right]+\beta\operatorname{Cov}\left[a,b\right]
\textrm{.}
$$
The equality (2) follows from the definition of Covariance and some manipulations:
$$
\operatorname{Cov}\left[{a},{c}\right]
=
\operatorname{Cov}\left[{a},\alpha{a}+\beta{b}\right]$$
$$=
\mathbb{E}\left[\left(a-\mathbb{E}\left[a\right]\right)\left(\alpha{a}+\beta{b} - \mathbb{E}\left[\alpha{a}+\beta{b}\right]\right)\right]$$ $$
=
\mathbb{E}\left[\left(\alpha{a^{2}}-2\alpha{a}\mathbb{E}\left[a\right]+\alpha\mathbb{E}\left[a\right]^{2}\right) + \left(\beta{ab}-\beta{a\mathbb{E}\left[b\right]}-\beta{b}\mathbb{E}\left[a\right]+\beta\mathbb{E}\left[a\right]\mathbb{E}\left[b\right]\right)\right]$$
$$
=
\mathbb{E}\left[\alpha\left(a-\mathbb{E}\left[a\right]\right)^{2} + \beta\left(a-\mathbb{E}\left[a\right]\right)\left(b-\mathbb{E}\left[b\right]\right)\right]
=
\alpha\operatorname{Var}\left[{a}\right] + \beta\operatorname{Cov}\left[a,b\right]
$$
You can get the Correlation from the Covariance in the obvious way, devide the Covariance by the square root of the product of the Variances of both random variables.
Hope I didn't miscalculate and that is the answer you were looking for.
