2024 NYJC Promo Q1

A curve \(C\) has equation

\[y = \frac{a}{x^2} + b\mathrm{e}^{2x} + c\,,\]

where \(a\), \(b\) and \(c\) are constants.

It is given that \(C\) passes through the point with coordinates \(\left(1, 2\mathrm{e}^2 - 1\right)\). When \(C\) undergoes a scaling of factor 2 parallel to the \(x\)-axis, the transformed curve passes through the point with coordinates \(\left(-1, 8 + \frac{2}{\mathrm{e}}\right)\). When \(C\) undergoes a translation by negative 1 unit in the direction of the \(y\)-axis, the transformed curve passes through the point with coordinates \(\left(\sqrt{3}, 2\mathrm{e}^{2\sqrt{3}} - 4\right)\). Find the values of \(a\), \(b\) and \(c\).[4]