Given that major axis is on x-axis, then the equation of ellipse is:

      x² /a² + y² /b² = 1 

=> 4² /a² + 3² /b² = 1

=> 16/a² + 9/b² = 1

=> 16b² + 9a² = a² *b²      ...........1

And

x² /a² + y² /b² = 1 

=> 6² /a² + 2² /b² = 1

=> 36/a² + 4/b² = 1

=> 36b² + 4a² = a² *b² .......2

Multiply by 9 in equation 2, we get

324b² + 36a² = 9a² *b² .......3

Subtract equation 1 - equation 3, we get

-260b² = -5a² *b² 

=> a² = 260/5

=> a² = 52

From eqaution 2, we get

36b² + 4(52) = (52) *b²      

=> 36b² + 208 = 52b²      

=> 52b² + 36b²  = 208

=> 16b²  = 208

=>b²  = 208/16

=> b²  = 13

So, the required equation is x² /52 + y² /13 = 1