For arbitrary $d \in \mathbb{R}^n$, I am interested in the set of $x \in \mathbb{R}^n$ such that $d \cdot \langle x_1^2, \dots, x_n^2\rangle = 0$. If $d > 0$, then $x = 0$ is the unique solution. If $d \ge 0$, then the solution set is a $k$-dimensional flat where $x_i$ can be anything iff $d_i = 0$ (otherwise, $x_i = 0$).
What about when $d$ contains both positive and negative elements? I suspect it's a hyperplane, on the observation that if $x$ is a solution, then $\lambda x$ is a solution for any $\lambda \in \mathbb{R}$. But does the figure always have enough dimension to be a hyperplane, or is it sometimes lower-dimensional?