Let $(X_1,X_2)$ be jointly normal with density
$$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$
Find unit vector $a=(a_1,a_2)$ so that
$$Var(a_1 X_1 + a_2 X_2) \ge Var(b_1 X_1 + b_2 X_2)$$ for all unit vectors $b = (b_1,b_2)$.
After some algebra I'm at this point
$$\begin{pmatrix} a_1\\ a_2 \end{pmatrix}^T\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix} \begin{pmatrix} a_1\\ a_2 \end{pmatrix} \ge \begin{pmatrix} b_1\\ b_2 \end{pmatrix}^T\begin{pmatrix} 1 & \rho\\ \rho & 1 \end{pmatrix} \begin{pmatrix} b_1\\ b_2 \end{pmatrix} $$
But this has been all algebra to me. What is the geometric intuition here? And how do I proceed to find this $(a_1, a_2)$?