# Level curves

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Describe the level curves of the function. Sketch the level curves for the given values of c.

f(x,y) = x^2 + 2y^2, c = 0,1,2,3,4

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#### Solution Preview

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 ...

#### Solution Summary

The expert describes the level curves of the function. Sketch the level curves for the given values of c.

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