Explanation

(b)
The centre and radius of given circle are $C_{1}\left( - \frac{3}{2},3 \right)$ and $r_{1} = \frac{9}{2}$
Let the centre and radius of required circle are $C_{2}(g,\ f)$ and $r_{2} = 2$
Since, the required circle is rolled outside the given circle.
$\therefore\ \ C_{1}C_{2} = r_{1} + r_{2}$
$\Rightarrow \ \ \sqrt{\left( g + \frac{3}{2} \right)^{2} + (f - 3)^{2}} = 2 + \frac{9}{2}$
$\Rightarrow \ \ g^{2} + \frac{9}{4} + 3g + f^{2} + 9 - 6f = \left( \frac{13}{2} \right)^{2}$
$\Rightarrow g^{2} + f^{2} + 3g - 6f = 31$
Hence, locus of the centre is
$x^{2} + y^{2} + 3x - 6y - 31 = 0$