Orthogonal trajectories?

**painterchica16** · October 5th 2008, 06:26 PM

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.
Thanks in advance!

**shawsend** · October 6th 2008, 07:33 AM

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:

$2x-2y_2\frac{dy_2}{dx}=0$

$\frac{dy_2}{dx}=\frac{x}{y}\Rightarrow \frac{dy_1}{dx}=-\frac{y}{x}$

or:

$y_1=\frac{C}{x}$

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