Answer
$$\frac{36}{89 \sqrt{89}}$$
Work Step by Step
By implicit differentiation:
$0=-8 x+(d y / d x)2 y$
$4 x / y=\frac{d y}{d x}$
$\frac{d^{2} y}{d x^{2}}=4 \cdot \frac{1 \cdot y-(d y / d x) \cdot x}{y^{2}}=4 \cdot \frac{y-(4 x / y) \cdot x}{y^{2}}=4 \cdot \frac{-4 x^{2}}{y^{3}+y^{2}}=\frac{36}{y^{3}}$
$\frac{36 /|y|^{3}}{\left(16 x^{2} / y^{2}+1\right)^{3 / 2}}=\kappa$
for $(x, y)=(2,5) \quad \kappa=\frac{36 / 125}{(64 / 25+1)^{3 / 2}}=\frac{36}{89 \sqrt{89}}$
