Angles made with the x – axis by two lines drawn through the point (1, 2) & cutting the line x + y = 4 at a distance $\sqrt{\frac{2}{3}}$ from the point $\left(1,2\right)$ are 

Let P = (1, 2) 
By parametric form, the coordinates of the point at a distance of $\sqrt{\frac{2}{3}}$ is give by  $\left(\text{1+}\sqrt{\frac{\text{2}}{\text{3}}}\text{cos\hspace{0.17em}θ,\hspace{0.17em}\hspace{0.17em}2+}\sqrt{\frac{\text{2}}{\text{3}}}\text{sin\hspace{0.17em}θ}\right)$, this point lies on x + y = 4.
Hence  $\mathbf{3}\mathbf{+}\sqrt{\frac{2}{3}}(\mathbf{sin\theta }\mathbf{+}\mathbf{cos}\mathbf{\theta })\mathbf{=}\mathbf{4}$. , on squaring an b.s we get 
  ${\text{sin}}^{\text{2}}{\text{θ+cos}}^{\text{2}}\text{θ\hspace{0.17em}+\hspace{0.17em}2sin\hspace{0.17em}θ\hspace{0.17em}cos\hspace{0.17em}θ=}\frac{\text{3}}{\text{2}}$$\Rightarrow \text{\hspace{0.17em}sin\hspace{0.17em}2θ\hspace{0.17em}=\hspace{0.17em}}\frac{\text{1}}{\text{2}}$
 The possible values of $\text{θ}$ are  $\frac{\text{π}}{\text{12}}\text{,\hspace{0.17em}}\frac{\text{5π}}{\text{12}}$