Level curves

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Solution. We analyze it as follows.

(1) If c=0, then f(x,y)=c gives rise to
x^2 + 2y^2=0 <==>x=y=0
So, in this case, the curve of the function is just a point
O(0,0).

(2) If c=1, then f(x,y)=c gives rise to
x^2 + 2y^2=1 <==> x^2/1^2+y^2/[(1/2)^(1/2)]^2=1
which is an ellipse with x-intercept 1 and y-intercept (1/2)^(1/2), in other words, the intersection points with x-axis is (-1,0) and (1,0) and the ...