Finding eqtns for level curves

Question

Given the function g(x, y) = x² /(2x − y²) .

Give the equations of the level curves of g at k = −1, 0, 1.

Answer 1

at k=1 (what is the z axis)

isn't curve a circle in the xy plane at z=1, (x-1)²+y²=1

k=0, x=0, y any number

and k=-1, (x+1)²-y²=1 is a hyperbola in xy plane at z=-1

Answer 2

Hi Nina L.

Have you looked up the definition of level curves? It's fairly simple. That value of k is a contour on which the value of the function (g) is fixed. So set each, successively, of the "k" values in for g(x,y), and solve the equation (that's 3 separate solutions, one for each "k"). The variables separate nicely, you will find. Don't forget that squareroot takes positive and negative values! What shape(s) do the curve have (within their respective g planes)? Can you figure out how these curves morph into each other as k continuously varies along the g axis? I.e., they form a 3-D shape in (x, y, g) space.

-- Cheers, --Mr. d.