Option 2 : \(\frac{{7\; - \;5\sqrt 7 }}{{1\; + \;\sqrt 7 }}\)

\({\rm{p}} = {\rm{}}\sqrt {44 - 16\sqrt 7 } {\rm{\;and\;q\;}} = {\rm{}}\sqrt {72 - 18\sqrt 7 } \)

⇒ \(p = \sqrt {(16 + 28 - 2\left( 4 \right)\left( {2\sqrt 7 } \right)}\)

⇒ \(p = \sqrt{\left( { 2\sqrt 7-4 } \right)^2}\)

⇒ p = 2√7 – 4

And \(q = \sqrt {(9 + 63 - 2\left( 3 \right)\left( {3\sqrt 7 } \right)}\)

⇒ \(q = \sqrt {{{\left[ {\;{3^2} + {{\left( {3\sqrt 7 } \right)}^2} - 2\left( 3 \right)\left( {3\sqrt 7 } \right)} \right]}^2}}\)

⇒ q = \(\sqrt {{{\left( { 3\sqrt 7 -3} \right)}^2}}\)

⇒ q = 3√7 – 3

∴ p + q = 2√7 – 4 + 3√7 – 3 = 5√7 – 7

p – q = 2√7 – 4 – 3√7 + 3 = – (1 + √7)

According to the question,

\(\therefore {\rm{\;}}\frac{{{\rm{p\;}} + {\rm{\;q}}}}{{{\rm{p\;}} - {\rm{\;q}}}} = \frac{{7{\rm{\;}} - {\rm{\;}}5\sqrt 7 }}{{1{\rm{\;}} + {\rm{\;}}\sqrt 7 }}\)