Answer
(a) $1.8$
(b) no
Work Step by Step
(a) $n_1sin45^{\circ}=n_2sin(\frac{45}{2})$
$\implies (1)(\frac{1}{\sqrt2})=n_2sin(22.5^{\circ})$
$\implies n_2=1.8$
(b) According to Snell's law
$n_1sin\theta_1=n_2sin\theta_2$
$\implies (1) sin45^{\circ}=nsin(0)$
$n=\frac{1}{\frac{\sqrt2}{0}}$
$n=\infty$
Hence the beams cannot meet at the center of the sphere.