I would solve this problem through the view of "Slope".
For a parabola y=a*x^2+b*x+c;
The tangential slope along the parabola line is a function of x.
slope=2*a*x+b; //the slope of reflection mirror
At the vertex point, the slope is 0.
For some point along the line, when light enters the parabola vertically , light is reflects to horizonal direction. The slopes of left point is -1 and right point is 1. That Y is the focus y;
So, to get the vertex X, the slope is 0. We solve the equation: 0=2*a*x+b;
We get x=-b/(2*a); // 125;
For the left refelction point, the slope is -1. We solve the equation -1=2*a*x+b;
We get x=(-1-b)/(2*a); //x1=110.02994011976
The focus Y=a*x*x+b*x+c; // we get 10.6100299401198;
If we want to test the right reflection point, then the slope is 1. We solve the equation: 1=2*a*x+b;
We get x=(1-b)/(2*a); //x2=139.97005988024;
The focus Y=a*x*x+b*x+c; // we get 10.6100299401197;
The focus Y calculated from left relection point and right reflection point are the same.
And the result is the same as calculated by a_slosh's method.
Many thanks to everyone who replied. I'm trying to get circles to realistically bounce off of a parabola. That involves getting tangent lines, so ericlin's response was relevant to that also. I'll post again later with the result. Now that I've got that down, another question arose. The circles have known radii. Is there an easy way to find a relation between the y value of the circles and the given angle? (see attached file for an example) The angle is a radius to the circle and is perpindicular to a tangent line shared by the circle and the parabola. Knowing that would make it much simpler to detect collision between the two, since you just have to test two points. I guess it's a little far out there, but hopefully someone around here will nudge me in the right direction.
I'm pretty pleased with the result, but I still need a better means of hit testing them than the midpoint. Each individual blob is actually pushed away from the wall based on its distance beyond the boundary (velocity) and a parabola tangent gotten from its y value (angle).
I decided to mess around with more collision testing today and I've gotten a solution that I'm pretty pleased with. Check out the attached diagram for reference. I made a horizontal line through the center of the circle. Where it intersects the left side of the parabola, I drew a tangent line and then drew line L perpindicular to that. Then I found angle a between line L and the horizontal line and used that same angle to draw a radius on the circle. Point P, where that radius intersects the circle, is very close to the first point that would touch the parabola if they close enough together, so I used P for collison testing on the left side. I made another point on the same horizontal plane as P on the right side of the circle to test the right side of the parabola.
Anyway, the whole thing yielded a pretty good result. It's still a little sketchy, but it looks a lot better than before.