Check orthogonality of these curves

Question

How do you check orthogonality condition for these curves, I know only the answer that they are but I can't seem to move forward with the solution. I appreciate your help.

Answer 1

The first derivative of each provides the slope of each. If the slopes are negative reciprocal the lines are orthogonal.

Answer 2

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

Answer 3

If ∫ x*y = 0, then the functions are orthogonal.