What is the orthogonal trajectory of y^2 - x^2 = C ??
Using implicit differentiation, find the slope of the tangents
y'=x/y
The slopes of the normals are therefore:
y'=-y/x
Rearrange and integrate
∫y'/y = ∫-1/x
ln y = -ln x + C1
ln y = -k ln x
raise to the power of e:
y=C/x
This is the family of reciprocal curves, or rectangular hyperbolas.