how can I solve two equations with two unknown variable using matlab?

I want to solve two equations with two unknown variables I have two equations (-x)*(x1 - x) + (r - y)*(y1 - y) = 0, (x1 - x)^2 + (y1 - y)^2 = z^2, where, x1,y1,r and z are known values. The values can be used as the inputs to the program.

what I tried is,

    syms x y
eqns = [x^2 +y^2 - x*point_x - y*point_y + r*point_y - r*y == 0 ,...
 (point_x - x)^2 + (point_y - y)^2 == length_tang^2];
vars = [x y];
[a,b] = solve(eqns,vars);

but it takes too much time and then gives no values.

[John D'Errico]

You say that x and y are unknowns. x1, y1, z are knowns.

(-x)*(x1 - x) + (r - y)*(y1 - y) = 0,

Then what is r? It is not an unknown. It is not known. Something in the middle perhaps? I also see an r1. Is it related to r? Is it known? Unknown? Quasi-known?

[Avi]

sorry for that. I have edited the question. there is only one r and it is known.

Answer 1

[John D'Errico]

point_x = 2;
point_y = 4;
length_tang = 1
r = 5;

syms x y
E = [(-x)*(point_x - x) + (r - y)*(point_y - y) == 0, (point_x - x)^2 + (point_y - y)^2 == length_tang^2];
S = solve(E,x,y)
S = 
  struct with fields:

      x: [2×1 sym]
      y: [2×1 sym]

S.x
ans =
   2
 6/5

S.y
ans =
    5
 17/5

Too slow? It came back virtually instantaneously.

timeit(@() solve(E,x,y))
ans =
      0.19394

0.2 seconds is too slow?

Note that it is probably possible for SOME sets of inputs, there is no real valued solution at all.

[Avi]

Thank you your answer works, but is slow as I have to solve it 51*61. Is there any way to increase the speed of the solution?

[Torsten]

Yes, you can get the general solution for point_x, point_y and r defined as "syms" and then substitute the 51*61 values for which you want to evaluate this solution.

help subs

Best wishes

Torsten.

Answer 2

[Torsten]

http://www.wolframalpha.com/input/?i=Solve%5B%5B-x*(x1-x)%2B(r-y)*(y1-y)%3D0+%26+(x1-x)%5E2%2B(y1-y)%5E2-z%5E2%3D0%5D,%5Bx,y%5D%5D

Best wishes

Torsten.