Advanced Geometry by H.C. Rajpoot-2014

The eccentricity of a conic section is given by $e = \frac {\sin \beta}{\sin \alpha}, \ 0^\circ < \alpha < 90^\circ, \ 0^\circ \le \beta \le 90^\circ,$ where $\beta$ is the angle between the plane and the horizontal and $\alpha$ is the angle between the cone's slant generator and the horizontal.

For $\alpha = 45^\circ$ and $\beta = 30^\circ$ , $e = \dfrac {\sin 30^\circ}{\sin 45^\circ} = \dfrac {\frac 12}{\frac 1{\sqrt 2}} = \boxed {\dfrac 1{\sqrt 2}}$ .

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Reference and image: Wikipedia: Eccentricity (mathematics)