There exist a closed form for the answer. hence the solution is O(K).

and it is easy to find with Maxima:

d2(x1,y1,x2,y2):=(x2-x1)^2+(y2-y1)^2;
x1: 0; /* even simpler if we assume x1=0 */
a1: d2(x1,y1,x,0); a2: d2(x2,y2,x,0); d: d2(x1,y1,x2,y2); /* the triangle */
ca: (a1+a2-d)/(2*sqrt(a1*a2)); /* cos of the angle */
s:solve(diff(ca,x)=0,x),ratsimp; /* solutions have y2-y1 as the denominator, so */
s1:solve(diff(ca,x)=0,x),y2=y1,ratsimp; /* let us solve it in this special case */