Find Lamé coefficients for the following curves

Question

How do you find lame coefficients for the following curves. I've been trying without getting any success. Would highly appreciate your help.

Answer 1

this is what I see

u(x,y)=x²-y²

v(x,y)=2xy

u+iv=z²,,,,where z=x+iy

z² is differentiable (meets du/dx=dv/dy) implying the intersection of u(x,y)=k and v(x,y)=k

is orthogonal.

to prove for this specific complex variable function

find x when u(x,y)=v(x,y)

and x when u_x=-1/v_x

for both 4x⁴-4kx²-k²=0