The correct option is (3) \(x={ \pm{A\over\sqrt2}}\)
Explaination::
potential energy =\({1\over2}kx^2={1\over2}m\omega^2( x^2)\)
Kinetic energy =\({1\over2}m(\omega\sqrt{ A^2-x^2})^2\)
As ,
potential energy = kinetic energy
\({1\over2}m\omega^2( x^2)={1\over2}m(\omega\sqrt{A^2-x^2})^2\)
\({1\over2}m\omega^2( x^2)={1\over2}m\omega^2( A^2)-{1\over2}m\omega^2( x^2)\)
\(m\omega^2( x^2)={1\over2}m\omega^2A^2\)
\(x^2={1\over2}A^2\)
\( x=\pm{A\over\sqrt2}\)