sketching the level curve of f(x,y) = y/(x^2 + y^2)

Just wondering if there is a way to get the equation of the line cause I have no clue if im supposed to do it another way or im just rusty and need to review my precalc notes.

yeah u need to review them omg i had a crush on this guy named bryan and he spelled it the same way as you! weird right?

To sketch the level curves of the function f(x, y) = y/(x^2 + y^2), you need to find the values of x and y for which f(x, y) is constant. Level curves represent contours of equal function values.

To find the equations of the level curves, you can follow these steps:

1. Set f(x, y) equal to a constant c: y/(x^2 + y^2) = c.

2. Multiply both sides of the equation by x^2 + y^2 to eliminate the denominator: y = c(x^2 + y^2).

3. Rearrange the equation to group y terms on one side: cy^2 - y + cx^2 = 0.

4. This is a quadratic equation in y. Solve for y using the quadratic formula:

y = [1 ± sqrt(1 - 4c²x²)] / (2c)

Note: The discriminant (1 - 4c²x²) must be non-negative for real solutions.

5. Once you have the equation for y in terms of x, you can plot points on the x-y plane by choosing specific values of c. Plotting multiple points for different values of c will give you a sense of how the curve varies as c changes.

6. Connect the plotted points to form the level curve. Repeat steps 4 and 5 for different values of c to sketch multiple level curves.

Remember that level curves are curves in two-dimensional space that represent the contours of equal function values. They can help visualize the behavior of the function f(x, y) = y/(x^2 + y^2) in relation to different values of c.