As shown in the figure, there are two distinct planes $\alpha$ and $\beta$ with intersection line containing two points $\mathrm{A}$ and $\mathrm{B}$ where $\overline{\mathrm{AB}} = 18$. A circle $C_1$ with diameter AB lies on plane $\alpha$, and an ellipse $C_2$ with major axis AB and foci $\mathrm{F}$ and $\mathrm{F'}$ lies on plane $\beta$. Let H be the foot of the perpendicular from a point P on circle $C_1$ to plane $\beta$. Given that $\overline{\mathrm{HF'}} < \overline{\mathrm{HF}}$ and $\angle\mathrm{HFF'} = \frac{\pi}{6}$. Let Q be the point on ellipse $C_2$ where line HF intersects it, closer to H, with $\overline{\mathrm{FH}} < \overline{\mathrm{FQ}}$. The circle on plane $\beta$ centered at H passing through Q has radius 4 and is tangent to line AB. If the angle between the two planes $\alpha$ and $\beta$ is $\theta$, what is the value of $\cos\theta$? (where point P is not on plane $\beta$) [4 points] (1) $\frac{2\sqrt{66}}{33}$ (2) $\frac{4\sqrt{69}}{69}$ (3) $\frac{\sqrt{2}}{3}$ (4) $\frac{4\sqrt{3}}{15}$ (5) $\frac{2\sqrt{78}}{39}$
As shown in the figure, there are two distinct planes $\alpha$ and $\beta$ with intersection line containing two points $\mathrm{A}$ and $\mathrm{B}$ where $\overline{\mathrm{AB}} = 18$. A circle $C_1$ with diameter AB lies on plane $\alpha$, and an ellipse $C_2$ with major axis AB and foci $\mathrm{F}$ and $\mathrm{F'}$ lies on plane $\beta$. Let H be the foot of the perpendicular from a point P on circle $C_1$ to plane $\beta$. Given that $\overline{\mathrm{HF'}} < \overline{\mathrm{HF}}$ and $\angle\mathrm{HFF'} = \frac{\pi}{6}$. Let Q be the point on ellipse $C_2$ where line HF intersects it, closer to H, with $\overline{\mathrm{FH}} < \overline{\mathrm{FQ}}$. The circle on plane $\beta$ centered at H passing through Q has radius 4 and is tangent to line AB. If the angle between the two planes $\alpha$ and $\beta$ is $\theta$, what is the value of $\cos\theta$? (where point P is not on plane $\beta$) [4 points]\\
(1) $\frac{2\sqrt{66}}{33}$\\
(2) $\frac{4\sqrt{69}}{69}$\\
(3) $\frac{\sqrt{2}}{3}$\\
(4) $\frac{4\sqrt{3}}{15}$\\
(5) $\frac{2\sqrt{78}}{39}$