Consider the surface given by

2x^2+5y^2+z^2=4.

The surface is sliced by planes of the form x=a, y=b or z=c for values of a, b, c given below. How do I match the resulting slices (in other words, the corresponding traces) with their correct plots?

- - Also the slices are:

1. z = 1.5

2. x = 1

3. z=0

4. y=0

This is just simple. For example you have a plane of the form x=a, then you just substitute x with a, and you'll get an equation with y and z only, hence you have a 2-d trace of the intersection. It is just similar for y=b and z=c.

(1) At z=1.5, 2x^2 + 5y^2 + 1.5^2 = 4

2x^2 + 5y^2 = 1.75

Now you have an ellipse in the z=1.5 plane as your trace.

(2) At x=1, 2 (1) ^2 + 5y^2 + z^2 = 4

5y^2 + z^2 = 2

Now you have an ellipse in the x=1 plane as your trace.

(3) At z=0, 2x^2 + 5y^2 + (0) ^2 = 4

2x^2 + 5y^2 = 4

Now you have an ellipse in the z=0 plane as your trace.

(4) At y=0, 2x^2 + 5 (0) ^2 + z^2 = 4

2x^2 + z^2 = 4

Now you have an ellipse in the y=0 plane as your trace.