Determine the equation of a plane tangent at a hyperboloid of one sheet
$\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{1}=1$
in a point M (2,3,1) . Prove that this tangent plane cuts the surface after two lines and find the angle between these two lines.
I was able to find the equation of the tangent plane by plugging in the coordinates of M:
$\frac{2x}{4}+\frac{3y}{9}-\frac{z}{1}=1$
which is equal to
$3x+4y-6z-6=0$
From here how can I prove that this plane cuts the surface after two lines?
Thank you a lot for any help given! I really appreciate it, because I'm a bit lost.
Best Answer
You made some mistake in those computations. Actually, the tangent plane is the plane$$x+\frac{2y}3-2z=2.$$And if you intersect this plane with the hyperboloid, you get the lines$$x=2\wedge y=3z\quad\text{and}\quad x=2z\wedge y=3.$$