Consider the hyperbola $\frac{x^{2}}{100} - \frac{y^{2}}{64} = 1$ with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle SPS_{1} = \alpha$, with $\alpha < \frac{\pi}{2}$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S_{1}P$ at $P_{1}$. Let $\delta$ be the distance of $P$ from the straight line $SP_{1}$, and $\beta = S_{1}P$. Then the greatest integer less than or equal to $\frac{\beta\delta}{9}\sin\frac{\alpha}{2}$ is ______.

Consider the hyperbola x²/100 - y²/64 = 1 with foci at S and S₁, where S lies on the positive x-axis. Let P be a point on the hyperbola, in the first quadrant. Let ∠ SPS₁ = α, with α < π/2. The straight line passing through the point S and having the same slope as that of the tangent at P to the hyperbola, intersects the straight line S₁P at P₁. Let δ be the distance of P from the straight line SP₁, and β = S₁P. Then the greatest integer less tha $\dfrac{x^2}{100}-\dfrac{y^2}{64}=1:\quad a=10,\ b=8,\ c=2\sqrt{41}$ $S(2\sqrt{41},0),\quad S_1(-2\sqrt{41},0),\quad P \text{ in first quadrant}$ $\angle SPS_1 = \alpha < \dfrac{\pi}{2}$ $\text{Line through } S \text{ with slope} = \text{tangent slope at } P$ $\text{meets } S_1P \text{ at } P_1;\quad \delta=\text{dist}(P,\ SP_1),\ \beta=S_1P$ $\text{Find } \left\lfloor \dfrac{\beta\delta}{9}\sin\dfrac{\alpha}{2} \right\rfloor$ $\text{Plan: } 1.\ \text{draw hyperbola, foci}$ $2.\ \text{tangent at } P;\ \text{line through } S \Rightarrow P_1$ $3.\ \text{use } P S_1 - PS = 2a \text{ and law of cosines}$ $\dfrac{x^2}{100}-\dfrac{y^2}{64}=1,\quad a=10,\ b=8$ $c=\sqrt{a^2+b^2}=\sqrt{164}=2\sqrt{41}$ $S(2\sqrt{41},0),\quad S_1(-2\sqrt{41},0)$ $P \text{ on the hyperbola, first quadrant}$ $\angle SPS_1=\alpha<\dfrac{\pi}{2}$ $\text{Tangent at } P(x_0,y_0):\ \dfrac{xx_0}{100}-\dfrac{yy_0}{64}=1$ $\text{slope } m=\dfrac{16x_0}{25y_0}$ $\text{through } S \text{ with slope } m:\quad SP_1 \parallel \text{tangent at } P$ $SP_1 \cap S_1P = P_1$ $\text{tangent at } P \text{ bisects } \angle SPS_1$ $\delta = \text{dist}(P,SP_1)=PS\cdot\sin\dfrac{\alpha}{2}$ $\beta=S_1P$ $d_2-d_1=2a=20,\quad d_1=SP,\ d_2=S_1P$ $SS_1^2=d_1^2+d_2^2-2d_1d_2\cos\alpha,\quad SS_1=2c$ $4c^2=(d_2-d_1)^2+2d_1d_2(1-\cos\alpha)$ $4c^2=4a^2+4d_1d_2\sin^2\dfrac{\alpha}{2}$ $d_1d_2\sin^2\dfrac{\alpha}{2}=c^2-a^2=b^2=64$ $\beta\delta\sin\dfrac{\alpha}{2}=d_1d_2\sin^2\dfrac{\alpha}{2}=64$ $\dfrac{\beta\delta}{9}\sin\dfrac{\alpha}{2}=\dfrac{64}{9}\approx 7.11$ $\left\lfloor \dfrac{\beta\delta}{9}\sin\dfrac{\alpha}{2} \right\rfloor = 7$