1. Fundamental Properties of a Standard Hyperbola

To solve this problem effectively, we must first recall the key properties of a hyperbola in its standard form. A hyperbola centered at the origin $(0,0)$ with its transverse axis along the x-axis is represented by the equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Here, $a$ is the length of the semi-transverse axis, and $b$ is the length of the semi-conjugate axis.

The crucial properties relevant to this problem are:

• Foci: The two foci are located at $(\pm ae, 0)$, where $e$ is the eccentricity of the hyperbola.
• Directrices: The equations of the two directrices are $x = \pm \frac{a}{e}$. Importantly, the directrix $x = \frac{a}{e}$ corresponds to the focus $(ae, 0)$, and $x = -\frac{a}{e}$ corresponds to the focus $(-ae, 0)$.
• Eccentricity ($e$): For any hyperbola, the eccentricity $e$ is always greater than 1 ($e > 1$). It is related to $a$ and $b$ by the identity: $e^2 = 1 + \frac{b^2}{a^2}$
• Length of the Latus Rectum ($l$): The length of the latus rectum is given by the formula: $l = \frac{2b^2}{a}$ These definitions form the bedrock of our problem-solving strategy.

2. Analyzing the Given Information

We are given a hyperbola $\mathrm{H}: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$. This confirms it is a standard hyperbola centered at the origin with its transverse axis along the x-axis.

The problem provides us with specific information:

• One focus of the hyperbola is at $(\sqrt{10}, 0)$.
• The directrix corresponding to this focus is $x=\frac{9}{\sqrt{10}}$.
• We need to calculate the value of the expression $9\left(e^2+l\right)$, where $e$ is the eccentricity and $l$ is the length of the latus rectum of H.

Our objective is to use the given focus and directrix to determine the values of $a$ and $e$. Once these are found, we will use the eccentricity relation to find $b^2$, then calculate $l$, and finally substitute all values into the target expression.

3. Step-by-Step Solution

Step 3.1: Forming Equations from Focus and Directrix Information

The problem provides one focus and its corresponding directrix. We will use the standard formulas for these properties to establish a system of equations involving $a$ and $e$.

• Using the Focus: We are given that one focus is at $(\sqrt{10}, 0)$. For a standard hyperbola with its transverse axis along the x-axis, the foci are at $(\pm ae, 0)$. By comparing the given focus with the standard form, we can write: $ae = \sqrt{10} \quad \text{..... (Equation 1)}$ Explanation: We equate the given coordinate of the focus to the standard formula for the focus. This sets up our first algebraic relationship between $a$ and $e$.

• Using the Corresponding Directrix: We are given that the directrix corresponding to the focus $(\sqrt{10}, 0)$ is $x = \frac{9}{\sqrt{10}}$. For a standard hyperbola, the directrix corresponding to the focus $(ae, 0)$ is $x = \frac{a}{e}$. Equating the given directrix equation with its standard form: $\frac{a}{e} = \frac{9}{\sqrt{10}} \quad \text{..... (Equation 2)}$ Explanation: We equate the given directrix equation to the standard formula for the directrix. It is crucial to ensure that the focus and directrix used are indeed corresponding pairs, which is explicitly stated in the problem. This gives us our second algebraic relationship.

Step 3.2: Solving for the Semi-Transverse Axis ($a$) and Eccentricity ($e$)

Now we have a system of two equations with two unknowns, $a$ and $e$:

1. $ae = \sqrt{10}$
2. $\frac{a}{e} = \frac{9}{\sqrt{10}}$

To efficiently solve for $a$ and $e$, we can multiply Equation 1 by Equation 2. This strategy is chosen because the eccentricity term '$e$' will cancel out, allowing us to directly solve for $a^2$: $(ae) \times \left(\frac{a}{e}\right) = \sqrt{10} \times \frac{9}{\sqrt{10}}$ Explanation: Multiplying the left sides results in $a^2 \times (e/e) = a^2$. On the right side, the $\sqrt{10}$ terms cancel out. This provides a direct path to finding $a^2$. $a^2 = 9$ Since $a$ represents a physical length (the semi-transverse axis), it must be a positive value. Therefore: $a = 3$

Now that we have the value of $a$, we can substitute it back into either Equation 1 or Equation 2 to find $e$. Let's use Equation 1: $3e = \sqrt{10}$ $e = \frac{\sqrt{10}}{3}$ Explanation: We substitute the calculated value of $a$ into one of the original equations to solve for $e$. As a quick check, $\sqrt{10} \approx 3.16$, so $e \approx \frac{3.16}{3} \approx 1.05$. This value is indeed greater than 1, which is a fundamental property of the eccentricity of a hyperbola.

Step 3.3: Determining the Semi-Conjugate Axis Squared ($b^2$) using the Eccentricity Relationship

With $a$ and $e$ determined, our next step is to find $b^2$, which is required for calculating the length of the latus rectum ($l$). We use the fundamental identity relating $a$, $b$, and $e$ for a hyperbola: $e^2 = 1 + \frac{b^2}{a^2}$ Explanation: This identity is a core property of hyperbolas. It provides the crucial link between the dimensions of the hyperbola ($a$ and $b$) and its eccentricity ($e$).

We have $e = \frac{\sqrt{10}}{3}$, so we calculate $e^2$: $e^2 = \left(\frac{\sqrt{10}}{3}\right)^2 = \frac{10}{9}$ We also have $a = 3$, so $a^2 = 9$. Substitute these values into the eccentricity formula: $\frac{10}{9} = 1 + \frac{b^2}{9}$ Now, we solve for $b^2$: $\frac{10}{9} - 1 = \frac{b^2}{9}$ $\frac{10-9}{9} = \frac{b^2}{9}$ $\frac{1}{9} = \frac{b^2}{9}$ $b^2 = 1$ Explanation: We perform algebraic simplification to isolate $b^2$. Although $b$ represents a length, only $b^2$ is needed for the latus rectum formula, so we don't necessarily need to calculate $b$ itself.

Step 3.4: Calculating the Length of the Latus Rectum ($l$)

Now that we have $a$ and $b^2$, we can calculate the length of the latus rectum, $l$, using its formula: $l = \frac{2b^2}{a}$ Explanation: This formula directly relates the length of the latus rectum to the semi-axes lengths, which we have already calculated in previous steps.

Substitute the values $b^2 = 1$ and $a = 3$: $l = \frac{2 \times 1}{3}$ $l = \frac{2}{3}$

Step 3.5: Final Calculation of the Expression $9(e^2+l)$

The problem asks for the value of the expression $9(e^2+l)$. We have all the necessary components:

• $e^2 = \frac{10}{9}$ (from Step 3.3)
• $l = \frac{2}{3}$ (from Step 3.4)

Substitute these values into the expression: $9\left(e^2+l\right) = 9\left(\frac{10}{9} + \frac{2}{3}\right)$ Explanation: This is the final step where we combine all the previously calculated values into the expression required by the problem.

To simplify, we can distribute the 9 across the terms inside the parenthesis: $9\left(\frac{10}{9}\right) + 9\left(\frac{2}{3}\right)$ $= 10 + (3 \times 2)$ $= 10 + 6$ $= 16$

Thus, the value of $9(e^2+l)$ is $16$.

The correct option is (D).

4. Important Tips and Common Pitfalls

• Master Standard Forms: Always begin by correctly identifying the standard form of the conic section (hyperbola in this case) and recalling all its associated properties (foci, directrices, eccentricity, latus rectum). A common mistake is confusing hyperbola formulas with ellipse formulas, especially the eccentricity relation ($e^2 = 1 + \frac{b^2}{a^2}$ for hyperbola vs. $e^2 = 1 - \frac{b^2}{a^2}$ for ellipse).
• Corresponding Properties: Pay close attention to which focus corresponds to which directrix. For $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, the focus $(ae, 0)$ corresponds to the directrix $x = \frac{a}{e}$.
• Algebraic Precision: Be meticulous with your algebraic manipulations, especially when solving simultaneous equations and substituting values. Small errors can propagate and lead to an incorrect final answer.
• Consistency Checks: After finding $a, b, e$, quickly verify if they satisfy the basic properties of a hyperbola: $a$ and $b$ must be positive lengths, and $e$ must be greater than 1. This can help catch calculation errors early. For example, if $e$ turned out to be less than or equal to 1, it would indicate a calculation error, as this is impossible for a hyperbola.

5. Summary and Key Takeaway

This problem is a quintessential example of how to apply the fundamental definitions and relationships governing a hyperbola to determine its characteristics. The systematic approach involved:

1. Formulating Equations: Using the given focus and its corresponding directrix to set up a system of two simultaneous equations for the semi-transverse axis ($a$) and eccentricity ($e$).
2. Solving for Parameters: Solving these equations to find the numerical values of $a$ and $e$.
3. Utilizing Fundamental Identity: Employing the fundamental eccentricity relationship $e^2 = 1 + \frac{b^2}{a^2}$ to calculate the semi-conjugate axis squared ($b^2$).
4. Calculating Latus Rectum: Determining the length of the latus rectum ($l$) using its specific formula $l = \frac{2b^2}{a}$.
5. Final Evaluation: Substituting all calculated values into the target expression $9(e^2+l)$ to arrive at the final answer.

The key takeaway is that a strong understanding of the standard forms and properties of conic sections is paramount for solving such problems efficiently and accurately.