Answer to Question #146843 in Analytic Geometry for Sarita bartwal

Question

Identify the concoid 1+2x^2+9y^2=3z^2. Does the XY plane intersect with it? Justify your answer

Answer 1

It is well-known that the equation of a two-sheet hyperboloid is

$-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$

Since the equation $1+2x^2+9y^2=3z^2$ is equivalent to $-2x^2-9y^2+3z^2=1$, we conclude that the concoid $1+2x^2+9y^2=3z^2$ is a two-sheet hyperboloid.

If $z=0$, then the equation $2x^2+9y^2=-1$ has no real solutions, and therefore, the $XY$-plane does not intersect with the two-sheet hyperboloid 1+2x^2+9y^2=3z^2.