Official solution
Equation of ellipse is \[{{{x^2}} \over 4} + {{{y^2}} \over 3} = 1\]
Normal at P(2 cos \[\theta \], \[\sqrt 3 \sin \theta \]) is 2x sin\[\theta \] - \[\sqrt 3 y\,cos\theta \] = sin \[\theta \] cos \[\theta \] as the normal is parallel to 2x + y = 4
\[ \Rightarrow \] \[{2 \over {\sqrt 3 }}\tan \theta = - 2\]
\[ \Rightarrow \tan \theta = - \sqrt 3 \,\,......\,(i)\]
tangent at P(2 cos \[\theta \], \[\sqrt 3 \sin \theta \]) is
\[\sqrt 3 \] x cos \[\theta \] + 2y sin \[\theta \] = 2 \[\sqrt 3 \]
Passes through (4, 4)
\[ \Rightarrow \] 4\[\sqrt 3 \] cos \[\theta \] + 8 sin \[\theta \] = 2\[\sqrt 3 \] ....... (ii)
From (i) and (ii)
\[ \Rightarrow P\left( { - 1,{3 \over 2}} \right)\,\& \,Q(4,4)\]
\[ \Rightarrow PQ = \sqrt {25 + {{25} \over 4}} = {{5\sqrt 5 } \over 2}\]