- An ellipse has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1\) and eccentricity \(e _ { 1 }\) A hyperbola has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) and eccentricity \(e _ { 2 }\)
Given that \(e _ { 1 } \times e _ { 2 } = 1\)
- show that \(a ^ { 2 } = 3 b ^ { 2 }\)
Given also that the coordinates of the foci of the ellipse are the same as the coordinates of the foci of the hyperbola,
- determine the equation of the hyperbola.