JEE 2023 — Mathematics PYQ

So,
$\frac{\alpha -6}{6-\frac{9}{2}}=\frac{10-11}{11-\beta }=\frac{13-11}{11+8}$

$\Rightarrow \frac{2(\alpha -6)}{3}=\frac{-1}{11-\beta }=\frac{2}{19}$

$\Rightarrow \frac{2}{3}(\alpha -6)=\frac{2}{19}$

\begin{aligned}
 & \Rightarrow19\alpha-114=3\Rightarrow19\alpha=117 \\
 & \Rightarrow\alpha=\frac{117}{19} \\
 & \mathrm{And},\frac{-1}{11-\beta}=\frac{2}{19} \\
 & \Rightarrow-19=22-2\beta \\
 & \Rightarrow2\beta=41 \\
 & \Rightarrow\beta=\frac{41}{2} \\
 & \therefore\left(19\alpha-6\beta\right)^2=\left(19\times\frac{117}{19}-\frac{6\times41}{2}\right)^2 \\
 & =(117-123)^{2}=36
\end{aligned}