# Math Help - Orthogonal trajectories?

1. ## Orthogonal trajectories?

How do I show two equations are orthogonal?
x+y=0
x=sin y

and

How do I verify that the two families of curves are orthogonal when K and C are real numbers?
xy= C
x^2-y^2=K

These questions popped up on my hw and I'm kinda lost about how to do them.
2. Well, they're orthogonal at a point if $y'_1(p)=-\frac{1}{y'_2(p)}$ so for the first I'd write them as $y_1=-x$ and $y_2=\arcsin(x)$. Calculate derivatives and show at the origin the orthogonal requirement is met. For the second one:
$2x-2y_2\frac{dy_2}{dx}=0$
$\frac{dy_2}{dx}=\frac{x}{y}\Rightarrow \frac{dy_1}{dx}=-\frac{y}{x}$
$y_1=\frac{C}{x}$