1. Introduction to Hyperbola: Essential Definitions and Formulas

A hyperbola is a fascinating conic section defined by its unique geometric properties. For any point on the hyperbola, the ratio of its distance from a fixed point (the focus) to its distance from a fixed line (the directrix) is a constant value greater than 1. This constant ratio is called the eccentricity, denoted by $e$. For a hyperbola, it is always true that $e > 1$.

For a standard hyperbola centered at the origin, with its transverse axis (the axis containing the foci and vertices) lying along the x-axis, its equation is given by: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Here:

• $a$ is the length of the semitransverse axis, which is half the length of the transverse axis.
• $b$ is the length of the semi-conjugate axis, which is half the length of the conjugate axis (perpendicular to the transverse axis).

The key properties that connect $a$, $b$, and $e$ with the focus and directrix are:

• Foci: The coordinates of the foci are $(\pm ae, 0)$.
• Directrices: The equations of the directrices are $x = \pm \frac{a}{e}$.
• Fundamental Relationship: The semitransverse axis $a$, semi-conjugate axis $b$, and eccentricity $e$ are linked by the identity: $b^2 = a^2(e^2 - 1)$ This formula is crucial for hyperbolas because $e > 1$, ensuring that $e^2 - 1$ is positive and $b^2$ is well-defined.

Our goal is to calculate the value of $a^2 - b^2$ using the given information about the hyperbola's focus, directrix, and an equation involving its eccentricity.

2. Utilizing Given Focus and Directrix Information

Why this step? The problem provides specific geometric details (the focus and its corresponding directrix). By comparing these given values with the standard formulas for a hyperbola, we can establish direct algebraic relationships involving the unknown parameters $a$ (semitransverse axis) and $e$ (eccentricity). This allows us to set up a system of equations that we can solve to find these fundamental values.

We are given the following:

• The focus is $S(5, 0)$.
• The corresponding directrix is $5x = 9$.

1. From the Focus: For a standard hyperbola with its transverse axis along the x-axis, a focus is located at $(ae, 0)$ or $(-ae, 0)$. Given $S(5, 0)$, we can equate the x-coordinate: $ae = 5 \quad \text{(Equation 1)}$

2. From the Directrix: The given directrix is $5x = 9$, which can be rewritten as $x = \frac{9}{5}$. The directrix that corresponds to the focus $(ae, 0)$ is $x = \frac{a}{e}$. Equating the given directrix to the standard form: $\frac{a}{e} = \frac{9}{5} \quad \text{(Equation 2)}$

3. Determining the Semitransverse Axis ($a$) and Eccentricity ($e$)

Why this step? We now have a system of two algebraic equations with two unknowns ($a$ and $e$). Solving this system will yield the specific numerical values of $a$ and $e$ for this particular hyperbola. These values are foundational, as they are necessary for calculating $b^2$ and, ultimately, the desired expression $a^2 - b^2$.

Let's solve the system of equations:

1. $ae = 5$
2. $\frac{a}{e} = \frac{9}{5}$

• Solving for $a^2$: A strategic way to find $a^2$ is to multiply Equation 1 by Equation 2. This neatly eliminates $e$, allowing us to solve directly for $a^2$: $(ae) \times \left(\frac{a}{e}\right) = 5 \times \frac{9}{5}$ $a^2 = 9$ Since $a$ represents a physical length (the semitransverse axis), it must be a positive value. Therefore, $a = \sqrt{9} = 3$.

• Solving for $e$: Now that we have the value of $a$, we can substitute it back into either Equation 1 or Equation 2 to find $e$. Using Equation 1: $ae = 5$ $(3)e = 5$ $e = \frac{5}{3}$

  Tip: Always verify that the eccentricity $e$ satisfies the condition for a hyperbola, which is $e > 1$. In our case, $e = \frac{5}{3} \approx 1.67$, which is indeed greater than 1. This confirms our calculations are consistent with the nature of a hyperbola.

4. Verifying Eccentricity with the Given Quadratic Equation

Why this step? The problem provides an additional piece of information: an equation that the eccentricity $e$ must satisfy ($9e^2 - 18e + 5 = 0$). While we have already determined $e$ from the focus and directrix, substituting our calculated value of $e$ into this quadratic equation serves as an excellent self-check. If our $e$ value satisfies the equation, it confirms the consistency of all given data and the correctness of our calculations.

The given equation for eccentricity is: $9e^2 - 18e + 5 = 0$ Substitute our calculated value $e = \frac{5}{3}$: $9\left(\frac{5}{3}\right)^2 - 18\left(\frac{5}{3}\right) + 5$ $= 9\left(\frac{25}{9}\right) - (6 \times 5) + 5$ $= 25 - 30 + 5$ $= 0$ Since substituting $e = \frac{5}{3}$ into the equation results in $0 = 0$, our calculated eccentricity is consistent with all the given conditions.

Common Mistake: Sometimes, a problem might provide information that leads to multiple possible values for a parameter (e.g., if we solved the quadratic equation directly, we might get two values for $e$). It's important to use all given information (like the focus/directrix) to uniquely determine the correct value or to verify which value is consistent.

5. Calculating the Semi-conjugate Axis Squared ($b^2$)

Why this step? To find the desired expression $a^2 - b^2$, we need the value of $b^2$. The fundamental relationship $b^2 = a^2(e^2 - 1)$ is precisely what connects $a$, $e$, and $b$. Having already found $a$ and $e$, we can now directly calculate $b^2$.

Using the fundamental identity for a hyperbola: $b^2 = a^2(e^2 - 1)$ Substitute the values we found: $a = 3$ (so $a^2 = 9$) and $e = \frac{5}{3}$: $b^2 = (3)^2 \left( \left(\frac{5}{3}\right)^2 - 1 \right)$ $b^2 = 9 \left( \frac{25}{9} - 1 \right)$ To simplify the term inside the parenthesis, find a common denominator: $b^2 = 9 \left( \frac{25}{9} - \frac{9}{9} \right)$ $b^2 = 9 \left( \frac{25 - 9}{9} \right)$ $b^2 = 9 \left( \frac{16}{9} \right)$ Now, cancel out the 9 in the numerator and denominator: $b^2 = 16$

6. Final Calculation: $a^2 - b^2$

Why this step? This is the ultimate objective of the problem. We have successfully determined both $a^2$ and $b^2$ from the given information.

We have:

• $a^2 = 9$ (from Section 3)
• $b^2 = 16$ (from Section 5)

Now, we can calculate the required expression: $a^2 - b^2 = 9 - 16$ $a^2 - b^2 = -7$

Therefore, the value of $a^2 - b^2$ is $-7$.

7. Summary and Key Takeaways

This problem effectively tests our understanding of the fundamental properties of a hyperbola. The solution involved:

1. Relating given geometric information (focus and directrix) to standard formulas to form a system of equations for $a$ and $e$.
2. Solving for $a$ and $e$ from this system.
3. Verifying the calculated eccentricity with an additional condition provided in the problem, ensuring consistency.
4. Utilizing the fundamental identity $b^2 = a^2(e^2 - 1)$ to find $b^2$.
5. Performing the final calculation $a^2 - b^2$.

The ability to connect different pieces of information (focus, directrix, eccentricity relation) through standard formulas is crucial for solving conic section problems. Always remember to check for consistency, especially that $e > 1$ for a hyperbola.

The final answer is $\boxed{-7}$.