Two Reflection Eccentricities

1. The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has right focus $F$; a line through $F$ meets the ellipse at $A, B$, and $C$ is the reflection of $A$ across the origin $O$. If $CF \perp AB$ and $CF = AB$, find the eccentricity. 2. The hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a, b > 0$) has right focus $F$; a line through $F$ meets the hyperbola at $A, B$, and $C$ is the reflection of $A$ across $O$. If $CF \perp AB$ and $CF = FB$, find the eccentricity.