Finding orthogonal trajectories to the parabola y=kx²

[Delber]

Homework Statement

For a any parabola with the equation $$y=kx^{2}$$
I'm trying to find a curve that intersect every point of the parabola at right angles.

Homework Equations

For a perpendicular intersection the slope is $$-\frac{1}{m}$$

The Attempt at a Solution

I took the derivative and then took the negative reciprocal of the derivative.

$$\frac{dy}{dx} = -\frac{1}{2kx}$$

Then I isolated the variables on different sides and then integrated. I ended up with:

$$y+ \frac{1}{2k}*ln(|x|) = 0$$

My problem is when I graph the two functions there is only one intersection and I was wondering if there was any flaws in my logic I used to reach my answer.

 

[protonchain]

Don't forget that you get a constant +C when you integrate.

so

$$y+ \frac{1}{2k}*ln(|x|) = 0$$

would actually be

$$y+ \frac{1}{2k}*ln(|x|) = C$$