I have a question asking me to describe the level curves for the following functions f(x,y) = x^2 + 2y^2 - 3x= 0

and

f(x,y) = -6

How do I go about doing this, for the second case, would the level curve be just a line or an empty set?

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- Jan 31st 2010, 08:27 AMPenguin91Level Curves question
I have a question asking me to describe the level curves for the following functions f(x,y) = x^2 + 2y^2 - 3x= 0

and

f(x,y) = -6

How do I go about doing this, for the second case, would the level curve be just a line or an empty set? - Jan 31st 2010, 08:45 AMTheEmptySet
The level curves are the sets where the functions values are constant.

So for the first one set

$\displaystyle f(x,y)=c \iff x^2+2y^2-3x=c $

Now complete the square on the x terms to get

$\displaystyle \left(x-\frac{3}{2} \right)^2+2y=c+\frac{9}{2}$

$\displaystyle \left(x-\frac{3}{2} \right)^2+2y=\frac{2c+9}{2}$

$\displaystyle \frac{\left(x-\frac{3}{2} \right)^2}{\frac{2}{2c+9}}+\frac{y}{2c+9}=1$

This is a familiy of ellipses.

For the 2nd one think of the definition. Where is the function constant - Jan 31st 2010, 09:00 AMPenguin91
Thanks for your reply, that made more sense!