Need help in drawing a level curve.

AndreTheGiant

1. The problem statement, all variables and given/known data

i want to draw a level curve for the following but im having some trouble.

x-y^2/x^2+y


2. Relevant equations



3. The attempt at a solution

I know i'm supposed to set c = x-y^2/x^2+y, but i can't find a way to set y in terms of x and c.

Mark44

Let me see if I have this straight. Your function is f(x, y) = x - (y²/x²) + y, right? That's what you wrote.

AndreTheGiant

Whoops, guess I should've added brackets.

[x-y^2]/[x^2+y]

sorry bout that.

Mark44

Much better.

Your equation is z = (x - y²)/(x² + y)

The level curves are curves in some plane, z = c.

For example, in the plane z = 1, the level curve is the equation (x - y²)/(x² + y) = 1. Keep in mind that y can't equal -x².

Multiply both sides by x² + y. What equation do you get?

AndreTheGiant

c(x^2+y) = (x-y^2)?

Mark44

So if you move all of the terms to one side, what do you get?

AndreTheGiant

cx^2 + cy - x + y^2 = 0

so when c = 0 i get a parabola, but what about when c = 2 for example..?

SammyS

Complete the squares.

2x² - x + y² +2y = 0

2(x² - (1/2)x) + y² +2y = 0

2(x² - 2 (1/4)x + (1/4)² -1/16 ) + y² +2y + 1 - 1 = 0

2(x - 1/4)² + (y - 1)² = 1 + 1/8

(x - 1/4)²/(9/16) + (y - 1)²/(9/8) = 1

Do similar for c in general.