Question

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 $\angle \operatorname{SPS}_1=\alpha$, 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 ______

Answer 1

Correct Answer: 7

Given: The equation \( SP - PP = 20 \), where \( SP \) and \( PP \) are distances, and the relationship involving the variable \( \beta \), \( \delta \), and \( \alpha \).

Step 1: Equation 1

The first equation is:

\(\beta - \frac{\delta}{\sin \frac{\alpha}{2}} = 20\)

Step 2: Simplification of the equation

Next, we solve the following equation for \( \beta \) and \( \delta \):

\(\beta^2 + \delta^2 = 400 \Rightarrow 2\beta\delta \sin \frac{\alpha}{2}\)

Step 3: Relationship between \( SP \) and \( \delta \)

We are given:

\(\frac{1}{SP} = \sin \delta\)

Step 4: Calculating \( \cos \alpha \)

We now calculate \( \cos \alpha \) using the formula:

\(\cos \alpha = \frac{SP^2 + \beta^2 - 656}{2\beta \sin \frac{\alpha}{2}} \Rightarrow \cos \alpha = \frac{2\beta \sin \frac{\alpha}{2} - 256}{2\beta \sin \frac{\alpha}{2}} \Rightarrow \cos \alpha = \frac{2\beta \sin \frac{\alpha}{2} - 256}{2 \beta \sin \frac{\alpha}{2}}\)

Step 5: Applying the relation between \( \lambda \) and \( \alpha \)

Now, using the equation \( \lambda = 128 \), we get the following relationship:

\(\lambda (1 - \cos \alpha) = 128 \Rightarrow \beta \sin \frac{\alpha}{2} \cdot 2 \sin^2 \frac{\alpha}{2} = 128\)

Step 6: Solving for \( \beta \) and \( \alpha \)

We substitute the values:

\(\frac{\beta}{9} \sin \frac{\alpha}{2} \cdot 2 \sin^2 \frac{\alpha}{2} = 128 \Rightarrow \frac{\beta}{9} \sin \frac{\alpha}{2} = 64/9\)

From this, we find:

\(\left[ \frac{\beta}{9} \sin \frac{\alpha}{2} \right] = 7 \, \text{where} \, [ \cdot ] \, \text{denotes the greatest integer function.}\)

Conclusion

Therefore, the final result is \( \boxed{7} \).

Final Answer: The value of \( \left[ \frac{\beta}{9} \sin \frac{\alpha}{2} \right] \) is 7, as required by the problem.