b
(b) \(\frac{K}{E} = \frac{{\frac{1}{2}m{\omega ^2}({a^2} - {y^2})}}{{\frac{1}{2}m{\omega ^2}{a^2}}}\)

\(= \frac{{{a^2} - {y^2}}}{{{a^2}}} \)

\(= 1 - \frac{{{y^2}}}{{{a^2}}}\)

So, \(\frac{{\left( {\frac{{3E}}{4}} \right)}}{E} = 1 - \frac{{{y^2}}}{{{a^2}}}\)

\(\Rightarrow \frac{{{y^2}}}{{{a^2}}} = 1 - \frac{3}{4} = \frac{1}{4}\)

\( \Rightarrow y = \frac{a}{2}\).