Given, $\,\,\,\, y^{2}=a x^{2}+b^{2} \,\,\,\, ...(i)$
$\therefore \,\,2 y \frac{d y}{d x}=2 a x$
$\Rightarrow \,\,\,\frac{d y}{d x}=\frac{a x}{y}$
$\therefore $ Slope at $(2,3)=\left(\frac{d y}{d x}\right)_{(2,3)}=\frac{2 a}{3}$
But slope of given tangent is $m=4$
$\therefore \frac{2 a}{3}=4 \Rightarrow a=6$
Since, point $(2,3)$ lies on the curve so, it satisfies the equation of the curve
$\therefore \,\,\,(3)^{2}=6(2)^{2}+b$
$\Rightarrow \,\,\,b=-15$