7. Let $P$ be an endpoint of a chord passing through one focus of an ellipse and perpendicular to the focal axis. If the normal line at point $P$ passes through one endpoint of the minor axis, find the eccentricity of the ellipse.
7. Let the equation of the ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and since $P$ is the endpoint of the latus rectum, by Corollary 2 of Property 13, the tangent line through $P$ must pass through the intersection point $K\left(\frac{a^{2}}{c}, 0\right)$ of the directrix and the focal axis. The coordinates of point $P$ are $\left(c, \frac{b^{2}}{a}\right)$, and the coordinates of $B^{\prime}$ are $(0,-b)$. Since $P B^{\prime} \perp P K$, we have $\frac{\frac{b^{2}}{a}-(-b)}{c}=-\frac{c-\frac{a^{2}}{c}}{\frac{b^{2}}{a}}$, which simplifies to $\frac{b^{4}}{a^{2}}+\frac{b^{3}}{a}=a^{2}-c^{2}=b^{2}$, or $\frac{b^{2}}{a^{2}}+\frac{b}{a}-1=0$. Since $\frac{b}{a}>0$, we have $\frac{b}{a}=\frac{1}{2}(\sqrt{5}-1)$. Therefore, $e=\sqrt{1-\left(\frac{b}{a}\right)^{2}}=\sqrt{\frac{1}{2}(\sqrt{5}-1)}$ is the required result.