
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^2y^2=K
These questions popped up on my hw and I'm kinda lost about how to do them.
Thanks in advance!

Well, they're orthogonal at a point if $\displaystyle y'_1(p)=\frac{1}{y'_2(p)}$ so for the first I'd write them as $\displaystyle y_1=x$ and $\displaystyle y_2=\arcsin(x)$. Calculate derivatives and show at the origin the orthogonal requirement is met. For the second one:
$\displaystyle 2x2y_2\frac{dy_2}{dx}=0$
$\displaystyle \frac{dy_2}{dx}=\frac{x}{y}\Rightarrow \frac{dy_1}{dx}=\frac{y}{x}$
or:
$\displaystyle y_1=\frac{C}{x}$