finding a point on a surface? the surface is an ellipsoid

Question

I have drawn the cross-sections of the surface $2(x-1)^2 + (y+2)^2 +z^2 = 2$ for the given planes, but am now asked to write down a point which is on the surface. I have no idea how to go about this, never learnt this nor have it in my textbook. Can anyone point me in the right direction?

Answer 1

A point on the surface is any such point $(x,y,z)$ for which $2(x-1)^2 + (y+2)^2 + z^2 = 2$, as that is the definition of the surface.

To find a single point on the surface, plug in a couple of values into your equation and see what you get. For example,

$(1,1,1)$ is not on the surface because $2(1-1)^2 + (1+2)^2+1^2=10\neq 2$

OK, let's see, maybe a smaller $y$ value will be better, so, is the point $(1,0,1)$ on the surface? Still no, what else should I change?

Answer 2

If the planes of your cross-sections are parallel to the coordinate planes (for example, the plane $x=1$) and you have fully described the size and location of the cross-section in each plane, you should be able to take one of the extreme points of a cross-section (such as the rightmost point) and figure out the coordinates of that point. (One of the coordinates is simply the equation of the plane itself.)