Let k=y/x=(h4^2-h2^2)/(h3^2-h1^2) (from above)
Let n=x*m*m=(h3^2-h1^2)/4         (from above)

Original equation:
(x-x')^2+(y-y')^2=(z-z')^2/m^2
(x-1 )^2+(y-0 )^2=(h1  )^2/m^2
(x-1 )^2+(xk-0)^2=(h1  )^2*x/n
(x-1 )^2+(xk-0)^2=(h1  )^2*x/n
multiplying it out:
x^2-2x+1+x^2*k^2 =x*h1^2/n
combining terms:
x^2*(1+k^2)-x(2+h1^2/n)+1=0

let a=1+k^2
let b=2+h1^2/n
x1=(b+sqrt(b^2-4a))/2a
x2=(b-sqrt(b^2-4a))/2a
y1=k*x1
y2=k*x2