- $x_1^2+4x_2^2+9x_3^2=1$
- $x_1^2+4x_2^2-9x_3^2=1$
- $-x_1^2-4x_2^2+9x_3^2=1$
I have to sketch these three surfaces and determine which are "bounded", which are "connected", and what the points closest to/furthest from $(0,0)$ are for each surface.
For each $q\,(\vec{x})=\lambda_1c_1^2+\lambda_2c_2^2+\lambda_3c_3^2$, it seems like I have to intuitively just know which of $\left\{c_1,c_2,c_3\right\}$ is closest to the origin and set that equal to zero to solve for the level curves (level sets?).
Is there a more explicit formula/algorithm for this within linear algebra (i.e. no Lagrange Multipliers or partial derivatives)?