# Level Curves question

• Jan 31st 2010, 09:27 AM
Penguin91
Level 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, 09:45 AM
TheEmptySet
Quote:

Originally Posted by Penguin91
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?

The level curves are the sets where the functions values are constant.

So for the first one set

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

Now complete the square on the x terms to get

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

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

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

This is a familiy of ellipses.

For the 2nd one think of the definition. Where is the function constant
• Jan 31st 2010, 10:00 AM
Penguin91