Prove that a necessary and sufficient condition on a norm in $\mathbb{R}^2$ that there exist an inner product satisfying the equation $\|x\|^2 = \langle x, x\rangle$ for all $x \in \mathbb{R}^2$ is that the locus of the equation $\|x\|=1$ be an ellipse
How can I prove the second part, showing that if $\|x\| = 1$ is an ellipse then $\|x\|^2 = \langle x, x\rangle$ ? Also I would be interested in a basis free proof.
Suppose $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is the equation of the ellipse. Let $\vec u=(u_1,u_2)$ and $\vec v=(v_1,v_2),$
define $\langle\cdot,\cdot\rangle:\mathbb R^2\times \mathbb R^2\to \mathbb R$ by $\langle u,v\rangle=\frac{u_1v_1}{a^2}+\frac{u_2v_2}{b^2}$, and check that
$\langle\cdot,\cdot \rangle$ defines an inner product that induces a norm with the required property.