Statement-I : If P, Q, R, S are the ends of latus recta of the hyperbola $\frac{{\mathrm{x}}^2}{16}-\frac{{\mathrm{y}}^2}{9}=1$then the area of rectangle PQRS is 45 square units.

Statement-II : The centre of the hyperbola $\frac{{\left(\mathrm{x}+\mathrm{y}+1\right)}^2}{3}-\frac{{\left(\mathrm{x}-\mathrm{y}-3\right)}^2}{6}=1$ lies in second quadrant. Which of above statements is true

$$\begin{gathered} Given hyperbola\frac{{\mathrm{x}}^2}{16}-\frac{{\mathrm{y}}^2}{9}=1 \\ where \mathrm{a}=4, \mathrm{b}=3 \mathrm{and} \mathrm{e}=\frac{5}{4} \\ Statement \mathrm{I}: \\ Given \mathrm{P},\mathrm{Q},\mathrm{R},\mathrm{S} \mathrm{are} \mathrm{ends} \mathrm{of} \mathrm{latus} \mathrm{recta} \mathrm{of} \mathrm{given} \mathrm{hyperbola} \\ \sin ce \mathrm{rectangle} \mathrm{formed} \mathrm{by} \mathrm{the} \mathrm{ends} \mathrm{of} \mathrm{latus} \mathrm{recta}, \mathrm{the}n \mathrm{length} of rec\tan gle \mathrm{will} \mathrm{be} \mathrm{distance} \mathrm{between} \mathrm{two} \mathrm{foci} \mathrm{and} \mathrm{the} \mathrm{breadth} \mathrm{will} \mathrm{be} \mathrm{equal} \mathrm{to} \mathrm{the} \mathrm{length} \mathrm{of} \mathrm{latusr}e\mathrm{ctum} \\ \mathrm{So}, \mathrm{length} \left(\mathrm{l}\right)=2\mathrm{ae}=10 \\ \mathrm{breadth} \left(\mathrm{b}\right)=\frac{2{\mathrm{b}}^2}{\mathrm{a}}=\frac{18}{4} \\ \therefore \mathrm{Area}=\mathrm{l}\times \mathrm{b}=45 \\ \mathrm{statement} \mathrm{I} \mathrm{is} \mathrm{true}. \\ \mathrm{Statement} \mathrm{II}: \\ \mathrm{centre} \mathrm{is} \mathrm{point} \mathrm{of} \mathrm{inter} \mathrm{section} \mathrm{of} \mathrm{x}-\mathrm{y}+1=0 \mathrm{and} \mathrm{x}-\mathrm{y}-3=0 \\ \therefore \mathrm{centre}=(1,-2)\in {\mathrm{Q}}_4 \\ \therefore \mathrm{Statement} \mathrm{II} \mathrm{false}. \end{gathered}$$