Solution of equations $\dfrac{1}{x} + \dfrac{1}{y} = 21 \text{ and } \dfrac{1}{x} - \dfrac{1}{y} + 9 = 0$ is :

1. $x = \dfrac{1}{6}, y = -\dfrac{1}{15}$

2. $x = -\dfrac{1}{6}, y = \dfrac{1}{15}$

3. $x = -\dfrac{1}{6}, y = -\dfrac{1}{15}$

4. $x = \dfrac{1}{6}, y = \dfrac{1}{15}$

Given,

Equations :

$\Rightarrow \dfrac{1}{x} + \dfrac{1}{y} = 21 …….(1) \\[1em] \Rightarrow \dfrac{1}{x} - \dfrac{1}{y} + 9 = 0 \\[1em] \Rightarrow \dfrac{1}{x} - \dfrac{1}{y} = -9 …….(2)$

Adding equations (1) and (2), we get :

$\Rightarrow \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{x} - \dfrac{1}{y} = 21 + (-9) \\[1em] \Rightarrow \dfrac{2}{x} = 12 \\[1em] \Rightarrow x = \dfrac{2}{12} = \dfrac{1}{6}.$

Substituting value of x in equation (1), we get :

$\Rightarrow \dfrac{1}{x} + \dfrac{1}{y} = 21 \\[1em] \Rightarrow \dfrac{1}{\dfrac{1}{6}} + \dfrac{1}{y} = 21 \\[1em] \Rightarrow 6 + \dfrac{1}{y} = 21 \\[1em] \Rightarrow \dfrac{1}{y} = 21 - 6 \\[1em] \Rightarrow \dfrac{1}{y} = 15 \\[1em] \Rightarrow y = \dfrac{1}{15}.$

Hence, Option 4 is the correct option.