Answer
$$\frac{3}{2 \sqrt{2}}$$
Work Step by Step
We find:
$\frac{\left|r^{2}+2\left(\frac{d r}{d \theta}\right)^{2}-r \frac{d^{2} r}{d \theta^{2}}\right|}{\left[r^{2}+\left(\frac{d r}{d \theta}\right)^{2}\right]^{3 / 2}}=\kappa$
$C A S:$
$\frac{2+\theta^{2}}{\left(1+\theta^{2}\right)^{3 / 2}}=\kappa(\theta)$
$\frac{3}{2 \sqrt{2}}=\kappa(1)$
