Solution :
The general equation of orthogonal hyperbola :
(x-p)*(y-q)=ab
where (p,q) is its asymptotes cross connect.
Passes through (4,2) means :
(4-p)*(2-q)=a*b [1]
Passes through (5,6) means :
(5-p)*(6-q)=a*b [2]
Passes through (6,-1) means :
(6-p)*(-1-q)=a*b [3]
from [1] and [2] :
(4-p)*(2-q)=(5-p)*(6-q)
8-2p-4q+pq=30-6p-5q+pq
4p+q=22 [4]
from [2] and [3] :
(5-p)*(6-q)=(6-p)*(-1-q)
30-6p-5q+pq=-6+p-6q+pq
7p-q=36 [5]
from [4] and [5] :
4p+q=22
7p-q=36
------- +
11p=58
p=58/11 [6]
q=22-4*58/11
q=(242-232)/11
q=10/11 [7]
from [1], [6], and [7] :
(4-58/11)*(2-10/11)=a*b
(44-58)/11*(22-10)/11=a*b
-14*12/121=a*b
a*b=-168/121 [8]
from [1] to [8] we get the final solution :
(x-58/11)*(y-10/11)=-168/121
or
(11x-58)*(11y-10)=-168