The normal to the hyperbola x 2 a 2 - y 2 9 = 1 at the point 8 , 3 3 on it passes through the point

Given,

hyperbola: $\frac{x^2}{a^2}-\frac{y^2}{9}=1$

it passes through $\left(8,3\sqrt{3}\right)$

$\because \frac{64}{a^2}-\frac{27}{9}=1\Rightarrow a^2=16$

Now, equation of normal to hyperbola in point form

$\because \frac{16x}{8}+\frac{9y}{3\sqrt{3}}=16+9$

$\Rightarrow 2x+\sqrt{3}y=25$

$\left(-1,9\sqrt{3}\right)$ satisfies the equation.