show that the plane z=h intersects the surface S in an ellipse

Question

Have the equation

4x^2 + 9y^2 - z = 0

I'm asked to identify the type of surface. It's an Elliptic Paraboloid. I'm then asked to "Assuming h>0, show that the plane z=h intersects the surface S in an ellipse, and calculate the semi-axes of the ellipse." How am I meant to go about this?

Reshaping the equation gives me:

x^2/3^2 + y^2/2^2 = h/6^2

and that is about as far as I've gotten. Thanks

Answer 1

If $z=h>0$, the intersection will have equation

$$(\frac {2x}{\sqrt {h}})^2+(\frac {3y}{\sqrt {h}})^2=1$$

which represents an ellipse in the plane $z=h $.

thus the great axe is $a=\frac {\sqrt {h}}{2} $ and the small is $b=\frac {\sqrt {h}}{3} $.