$y^{2}=6 x ;$ slope of tangent at $\left(x_{1}, y_{1}\right)$ is $m_{1}=\frac{3}{y_{1}}$
also $9 x^{2}+b y^{2}=16 ;$ slope of tangent at $\left(x_{1}, y_{1}\right)$ is
$m_{2}=\frac{-9 x_{1}}{b y_{1}}$
As $m_{1} \,m_{2}=-1$
$\Rightarrow \frac{-27 x_{1}}{b y_{1}^{2}}=-1$
$\Rightarrow b=\frac{9}{2}\left(\right.$ as $\left.y_{1}^{2}=6 x_{1}\right)$