Ellipse : $${{{x^2}} \over {16}} + {{{y^2}} \over 7} = 1$$

Eccentricity $$ = \sqrt {1 - {7 \over {16}}} = {3 \over 4}$$

Foci $$ \equiv ( \pm \,a\,e,0) \equiv ( \pm \,3,0)$$

Hyperbola : $${{{x^2}} \over {\left( {{{144} \over {25}}} \right)}} - {{{y^2}} \over {\left( {{\alpha \over {25}}} \right)}} = 1$$

Eccentricity $$ = \sqrt {1 + {\alpha \over {144}}} = {1 \over {12}}\sqrt {144 + \alpha } $$

Foci $$ \equiv ( \pm \,a\,e,0) \equiv \left( { \pm \,{{12} \over 5}\,.\,{1 \over {12}}\sqrt {144 + \alpha } ,\,0} \right)$$

If foci coincide then $$3 = {1 \over 5}\sqrt {144 + \alpha } \Rightarrow \alpha = 81$$

Hence, hyperbola is $${{{x^2}} \over {{{\left( {{{12} \over 5}} \right)}^2}}} - {{{y^2}} \over {{{\left( {{9 \over 5}} \right)}^2}}} = 1$$

Length of latus rectum $$ = 2\,.\,{{{{81} \over {25}}} \over {{{12} \over 5}}} = {{27} \over {10}}$$