Finding the intersection of a line and a parabola numerically

hello, I have come to a stall when trying to figure out how to obtain the two elements that meet a single condition. So if I have a parabolic function and a line(condition) going through it at two different points, how can I obtain the two numerical values for those points?

This is where I am stuck:

dd=logspace(-9,1,100);

uu=sqrt(A_n*(sig_cal*g*dd(1,:)+gamma./(1.225*dd(1,:))));

u2=dd+v_dw_ms;

plot(dd,uu,'r',dd,u2,'b')

where v_dw_ms is a certain constant value on the uu axis.

I can obtain the points graphically, but I need the actual numerical values for further numerical analysis.

Thanks in advance.

Answer 1

[Image Analyst]

Just set the line and parabola equal to each other and solve for x

yParabola = a*x^2 + b*x + c
yLine = m*x+d
a*x^2 + b*x + c = m*x + d
a*x^2 + (b-m)*x + (c-d) = 0
Then use the quadratic solution formula to solve for the 2 x:
x1 = (-(b-m)-sqrt((b-m)^2-4*a*(c-d))) / (2*a)
x2 = (-(b-m)+sqrt((b-m)^2-4*a*(c-d))) / (2*a)

[Luis Canales Tough]

That would work, but I think I made an error saying parabolic function, simply because the graph looks similar to a parabola.

The equation is actually of the form Y=sqrt(X + 1/X), so the quadratic formula does not apply.

[Image Analyst]

So simply set them equal like I said and then simplify the equation. I think you can then use roots() or fsolve() or something.