$($c$)($i$)$ In the Edgeworth Box diagram, explain why both consumers' Marginal Rate of Substitution

are equal at every point on the contract curve.

$($ii$)$ In a two person two commodity pure exchange model, the utility functions are

$U_1=q_{11}^{}{q}_{12}$ and $U_2=q_{21}{q}_{22}$ and given that $q_{11}+q_{21}=q_1$ and $q_{12}+q_{22}=q_2.$ Derive

the contract curve as an implicit function of $q_{11}$ and $q_{12}.$