Infer relationships between coefficients given covariance information Quant Interview Question:
Given $y$ = $a_1$$x_1$, $y$ = $b_1$$x_1$+$b_2$$x_2$, $cov(x_1,y)$ != 0, $cov(x_1,x_2)$ > 0, $cov(x_2,y)$ = 0. Compare $abs(a_1)$ and $abs(b_1)$. Which one is larger and why?
My intuition tells me that this has something to do with linear regression since covariance is given. The OLS coefficient estimates consist of the covariance, but after writing them out, I'm not sure how to proceed. Specifically, I'm not sure how to move towards the absolute of the coefficients and how to move from the estimates to the true values.
Any help would be greatly appreciated!

Using the bilinearity of the covariance, the definition of the variance of a random variable, and $\mathrm{Cov}\left(x_2, y\right)=0$, we get
$$
\begin{align}
0 \leq \mathbb V\left(y\right)
&= \mathrm{Cov}\left(y, y\right) \\
&= \mathrm{Cov}\left(b_1 x_1 + b_2 x_2, y\right) \\ 
&= \mathrm{Cov}\left(b_1 x_1, y\right) + \mathrm{Cov}\left(b_2 x_2, y\right) \\
&= b_1\mathrm{Cov}\left(x_1, y\right) + b_2\mathrm{Cov}\left(x_2, y\right) \\
&= b_1\mathrm{Cov}\left(x_1, a_1 x_1\right) + 0 \\
&= a_1b_1\mathbb V\left(x_1\right).
\end{align}
$$
From $\mathrm{Cov}\left(x_1, y\right) \neq 0$ we now see that $a_1 \neq 0,\mathbb V\left(x_1\right) \neq 0,$ and either $\mathrm{sgn}\left(a_1\right) = \mathrm{sgn}\left(b_1\right)$ or $b_1=0$.
We also have
$$
\mathbb V\left(y\right) = \mathbb V\left(a_1 x_1 \right) = a_1^2\mathbb V\left(x_1 \right)
$$
and thus
$$
a_1b_1\mathbb V\left(x_1\right)= a_1^2\mathbb V\left(x_1 \right) \iff a_1 b_1 = a_1^2 \iff b_1 = a_1.
$$
