Problem: Given that $L$ is a subspace of $\mathbb{R}^n$ and denote $L^\perp$ as the orthogonal complement of $L$, i.e. we can write $\mathbb{R}^n = L \bigotimes L^\perp$. Let $w \in \mathbb{R}^n$ be a fixed vector and $M \in \mathbb{R}^{n \times n}$ be positive definite. Find all vectors $x \in \mathbb{R}^n$ such that $$\begin{cases} x \in L \\ Mx - w \in L^\perp \end{cases} $$
When I approach this problem, I noticed that the two conditions imply the following quadratic equation $$ \langle x, Mx - w \rangle = 0 \Longleftrightarrow \langle x, Mx \rangle - \langle x, w \rangle = 0. $$ Hence, we can just find $x \in L$ such that this equation satisfies. However, I have no idea how to proceed from here.
On a further note, I figure that the set of points that satisfies this equation is kind of a "generalized parabola" to the $n$-dimensional situation, since $M$ is positive definite and $-\langle x, w \rangle$ is a linear term that got added to the positive definite quadratic form $\langle x, M \rangle$. Then we would find the intersection of such "generalized parabola" with the linear subspace $L$ to obtain all the points we need.
When $\mathbb{R}^n = \mathbb{R}^2$, the problem is really similar to finding the intersections of a parabola with a line $L$ (since the $0$-dimensional and $2$-dimensional cases of $L$ are trivial here, we consider $\dim L = 1$).
Thank you for your attention. Any hint would be greatly appreciated.