Explanation

(b)
Given circles are $x^{2} + y^{2} - y = 0$ and $x^{2} + y^{2} + y = 0$ centres and radii of these circles are $C_{1}\left( 0,\frac{1}{2} \right),\ C_{2}\left( 0,\ - \frac{1}{2} \right)$
And $r_{1} = \frac{1}{2},\ r_{2} = \frac{1}{2}$
Now, $C_{1}C_{2} = \sqrt{0 + \left( \frac{1}{2} + \frac{1}{2} \right)^{2}} = 1$
And $r_{1} + r_{2} = \frac{1}{2} + \frac{1}{2} = 1$
$\because\ \ \ C_{1}C_{2} = r_{1} + r_{2}$
It means that two circles touch each other externally
Hence, number of common tangents are 3