The centre of the ellipse _{$\frac{{\left(x+y-2\right)}^2}{16}+\frac{{\left(x-y\right)}^2}{9}=1$} is:

The equation of the ellipse is _{$9{\left(x+y-2\right)}^2+16{\left(x-y\right)}^2=144\to \left(1\right)$}

Differentiating (1) with respect to x, treating y as constant , we get , _{$18\left(x+y-2\right)+32\left(x-y\right)=0\to \left(2\right)$}

Differentiating (1) with respect to y, treating x as constant , we get ,

_{$18\left(x+y-2\right)-32\left(x-y\right)=0\to \left(3\right)$}

Solving (2) and (3), we get the coordinates of the centre (1, 1).