Relation Between Ka And Kb, Common Ion Effect

Weak acid is related is related to its conjugate base through equilibrium constant expression k_a
HA +H₂O ↔ H₃O⁺ +A^-

K_a = $\frac{\text{[}{{\text{H}}_{\text{3}}\text{O}}^{\text{+}}\text{][}{\text{A}}^{\text{-}}\text{]}}{\text{[HA]}}$                                                                                                                         

Weak base is related is related to its conjugate acid through equilibrium constant expression k_b
BOH + H₂O ↔ B⁺ + OH^-

K_b =  $\frac{\text{[B}{}^{\text{+}}\text{][}{\text{OH}}^{\text{-}}\text{]}}{\text{[BOH]}}$   

As the conjugated base pairs are related through the equilibrium constant we can assume that the equilibrium constant for an acid-base pair also related to each other.
Consider the dissociation of weak acid in water
HA +H₂O ↔ H₃O⁺ +A^-

 K_a = $\frac{\text{[}{{\text{H}}_{\text{3}}\text{O}}^{\text{+}}\text{][}{\text{A}}^{\text{-}}\text{]}}{\text{[HA]}}$     ...........(1)

HA being a weak acid it has strong conjugate base A^-
The dissociation equilibrium of strong conjugate base in water is
A^-  + H₂O ↔ HA +OH^-

 K_b = $\frac{\text{[HA][}{\text{A}}^{\text{-}}\text{]}}{\text{[}{\text{A}}^{\text{-}}\text{]}}$      ...........(2)

Multiplying equation 1 & 2

K_a * K_b  = $\frac{\text{[}{{\text{H}}_{\text{3}}\text{O}}^{\text{+}}\text{][}{\text{A}}^{\text{-}}\text{]}}{\text{[HA]}}$ * $\frac{\text{[HA][}{\text{A}}^{\text{-}}\text{]}}{\text{[}{\text{A}}^{\text{-}}\text{]}}$ 
             = [H₃O⁺ ][ OH^-] = k _w
 k _w  = Ionic product of water
∴  k_a * k_b = k _w

Common ion effect:  The common ion effect is defined as the shift in the position of an ionic equilibrium caused by the addition of a solute that provides an ion which is the part of equilibrium.
Consider the dissociation of acetic acid in water
The equilibrium is represented as 
CH₃COOH _(aq) ↔ CH₃COO^- +H₃O⁺
To this solution add a small amount of sodium acetate.
dissociation of sodium acetate is represented as
CH₃COONa → CH3COO^- +Na⁺
Since the concentration of CH3COO^-  ions increased in the solution ,according to Le Chatelier,s principle the equilibrium will be shifted to left.
Due to the presence of common ions CH₃COO^- the equilibrium will be shifted to the left side. And the dissociation of  CH₃COOH is suppressed.