prove that the equation of the chord of the hyperbola xy=c^2 which is bisected at the point (2c,3c) is 3x+2y=12c

This can be solved using transformation of curves. Equation of chord when mid point is given is by T=S₁ where T is the transformed equation and S₁ is the value of curve when point is put into it. Transformation for xy=(x₁y+xy₁)/2. So,

(2cy+3cx)/2 – c² = 6c² – c² . Hence, the answer is 3x+2y=12c.