The tangent at the point (t,1/t) on the hyperbola xy=1 meets the x and y axes at A and B respectively. The point C on the line AB is such that AC:AB=a:b. prove that the locus of C as t varies is the rectangular hyperbola

I know A(2t,0) and B(0,2/t). So i let C(x,y) and found the values of AC and BC, then used the ratio

I got

But now i don't know what to do next, Don't see any ways to make the given equation.