Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$. Let e' and l' respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If ${\mathrm{e}}^2=\frac{11}{14}\mathrm{l}\text{ and }{\left({\mathrm{e}}^{\text{'}}\right)}^2=\frac{11}{8}\mathrm{l}\text{'}$, then the value of 77a + 44b is equal to :

$\mathrm{e}=\sqrt{\frac{{\mathrm{a}}^2+{\mathrm{b}}^2}{{\mathrm{a}}^2}}, \mathrm{\ell }=\frac{2{\mathrm{b}}^2}{\mathrm{a}}$

Given ${\mathrm{e}}^2=\frac{11}{14}\mathrm{\ell }$

$\begin{array}{l}1+\frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}=\frac{11}{14}\cdot \frac{2{\mathrm{b}}^2}{\mathrm{a}}\\ \frac{{\mathrm{a}}^2+{\mathrm{b}}^2}{{\mathrm{a}}^2}=\frac{11}{7}\cdot \frac{{\mathrm{b}}^2}{\mathrm{a}} ....\left(1\right)\end{array}$

Also $\mathrm{e}\text{'}=\sqrt{\frac{{\mathrm{a}}^2+{\mathrm{b}}^2}{{\mathrm{b}}^2}}, \mathrm{\ell } \text{'}=\frac{2{\mathrm{a}}^2}{\mathrm{b}}$

Given ${\left(\mathrm{e}\text{'}\right)}^2=\frac{11}{8}\mathrm{\ell }\text{'}$

$\begin{array}{l}1+\frac{{\mathrm{a}}^2}{{\mathrm{b}}^2}=\frac{11}{8}\cdot \frac{2{\mathrm{a}}^2}{\mathrm{b}}\\ \frac{{\mathrm{a}}^2+{\mathrm{b}}^2}{{\mathrm{b}}^2}=\frac{11}{4}\cdot \frac{{\mathrm{a}}^2}{\mathrm{b}} ....\left(2\right)\\ \text{ New (1) }÷\left(2\right)\\ \frac{{\mathrm{b}}^2}{{\mathrm{a}}^2}=\frac{4}{7}\cdot \frac{{\mathrm{b}}^3}{{\mathrm{a}}^3}\\ \therefore 7\mathrm{a}=4\mathrm{b} ....\left(3\right)\\ \text{ From }\left(2\right)\\ \frac{16{\mathrm{b}}^2}{49}+{\mathrm{b}}^2\\ \frac{{\mathrm{b}}^2}{49}=\frac{11}{4}\cdot \frac{16{\mathrm{b}}^2}{49\mathrm{b}}\\ \frac{65}{49}=\frac{11}{4}\cdot \frac{16}{49}\cdot \mathrm{b}\\ \therefore \mathrm{b}=\frac{4\times 65}{11\times 16} ....\left(4\right)\end{array}$

Now the value of

$\begin{array}{l}77\mathrm{a}+44\mathrm{b}\\ 11(7\mathrm{a}+4\mathrm{b})=11(4\mathrm{b}+4\mathrm{b})=11\times 8\mathrm{b}\\ \therefore \text{ Value of }11\times 8\mathrm{b}=11\times 8\times \frac{4\times 65}{16\times 11}=130\end{array}$