The line $21\mathrm{x} + 5\mathrm{y} = \mathrm{\lambda }$ touches the hyperbola $7{\mathrm{x}}^2 – 5{\mathrm{y}}^2 = 232$ then $\mathrm{\lambda }=$

Given hyperbola is $\frac{{\mathrm{x}}^2}{{\displaystyle \frac{232}{7}}}-\frac{{\mathrm{y}}^2}{{\displaystyle \frac{232}{5}}}=1$

Given tangent $21\mathrm{x}+5\mathrm{y}-\mathrm{\lambda }=0$

$$\begin{gathered} Apply \tan gent condition n^2=a^2{l}^2-b^2{m}^2 \\ \Rightarrow {\mathrm{\lambda }}^2=\frac{232}{7}{\left(21\right)}^2-\frac{232}{5}{\left(5\right)}^2 \Rightarrow {\mathrm{\lambda }}^2=232[63-5] \\ \Rightarrow \mathrm{\lambda }=\sqrt{232\times 58} \\ \Rightarrow \mathrm{\lambda }=\sqrt{116\times 116} \\ \Rightarrow \mathrm{\lambda }=116 \end{gathered}$$