The circumcentre of the triangle formed by the lines ${\mathrm{x}}^2-{\mathrm{y}}^2=0$ and $\mathrm{y}-5=0$is 

The given lines are $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$   
Observing the lines, two sides are perpendicular lines, so that the triangle formed by the given lines is right angled triangle. 
For the right-angled triangle, the ortho center is the vertex at right angle, and the circum centre is midpoint of the hypotenuse. 
 The vertices of the triangle are $\left(0,0\right)\left(-5,5\right),\left(5,5\right)$
The origin is the vertex is at right angle, because it is the point of intersection of two perpendicular lines
The midpoint of the other two vertices is circum center, hence circum center is $\boxed{\left(0,5\right)}$