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Step-by-Step Solution
Step 1: Identify the given information
1. The hyperbola has a semitransverse axis $a$ and semi-conjugate axis $b$.
2. Its eccentricity $e$ satisfies the equation $9e^2 - 18e + 5 = 0$.
3. A focus is given at $S(5,0)$.
4. The corresponding directrix is $5x = 9$, or $x = \tfrac{9}{5}$.
Step 2: Use the focus relation for a hyperbola
For a rectangular coordinate system with center of the hyperbola at the origin, the focus at $(ae,0)$ tells us:
$ae = 5 \quad \dots (1)$
Step 3: Use the directrix property
For the same hyperbola, the corresponding directrix is given by $\tfrac{x}{a/e}=1$, which means the directrix is $x = \tfrac{a}{e}$. We know from the question that the directrix is $x = \tfrac{9}{5}$. Hence:
$\tfrac{a}{e} = \tfrac{9}{5} \quad \dots (2)$
Step 4: Solve for $a$ and $e$
From equations (1) and (2):
$ ae = 5 \quad \text{and} \quad \tfrac{a}{e} = \tfrac{9}{5}. $
Multiply these two equations:
$\bigl(ae\bigr)\Bigl(\tfrac{a}{e}\Bigr) = 5 \times \tfrac{9}{5} = 9.$
Thus, $a^2 = 9$, giving $a = 3$ (taking the positive value for the hyperbola's semitransverse axis). Substituting $a=3$ into $ae=5$ yields:
$3e = 5 \quad \Longrightarrow \quad e = \tfrac{5}{3}.$
Step 5: Find $b^2$ using the eccentricity relation
For a hyperbola with semitransverse axis $a$, eccentricity $e$, and semi-conjugate axis $b$, we have:
$b^2 = a^2 \bigl(e^2 - 1\bigr).
Substituting $a=3$ and $e = \tfrac{5}{3}$, we get:
$b^2 = 9 \Bigl(\Bigl(\tfrac{5}{3}\Bigr)^2 - 1\Bigr) = 9 \Bigl(\tfrac{25}{9} - 1\Bigr) = 9 \Bigl(\tfrac{25 - 9}{9}\Bigr) = 9 \times \tfrac{16}{9} = 16.
Step 6: Calculate $a^2 - b^2$
Now,
$a^2 - b^2 = 9 - 16 = -7.
Hence, the required value of $a^2 - b^2$ is $-7$.
