Step-by-step Solution:

Suppose the equation of the ellipse is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), since the ellipse passes through the points \((4,3)\) and ( \(-1,4\) ), thus \[\] \(\frac{16}{a^{2}}+\frac{9}{b^{2}}=1\) and \(\frac{1}{a^{2}}+\frac{16}{b^{2}}=1\) \[\] Suppose \(\frac{1}{a^{2}}=A\) and \(\frac{1}{b^{2}}=B\), then the equations become, \[\] \(16 A+9 B=1\) and \(A+16 B=1\) \[\] Solving these equations \(A=\frac{7}{247}\) and \(B=\frac{15}{247}\). \[\] Equation of the ellipse is \(7 x^{2}+15 y^{2}=247\). \[\] The same result can be obtained by putting the given points in the four choices\[\]Choice (B)\[\]