Key Concepts and Formulas

To solve this problem, we need to recall the standard properties of a hyperbola. Specifically, we will use:

1. Foci of a Hyperbola: For a hyperbola with its center at $(h, k)$ and its transverse axis parallel to the x-axis, the coordinates of the foci are given by $(h \pm ae, k)$. Here, $a$ is the length of the semi-transverse axis, and $e$ is the eccentricity.
2. Relationship between $a, b, e$: For a hyperbola, the lengths of the semi-transverse axis ($a$), semi-conjugate axis ($b$), and eccentricity ($e$) are related by the equation: $b^2 = a^2(e^2 - 1)$
3. Length of the Latus Rectum: The length of the latus rectum of a hyperbola is given by the formula: $L = \frac{2b^2}{a}$

Given Information:

• Foci of the hyperbola: $(1 \pm \sqrt{2}, 0)$
• Eccentricity of the hyperbola: $e = \sqrt{2}$

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Step 1: Determine the Center and the product $ae$ from the Foci

The given foci are $(1 \pm \sqrt{2}, 0)$. We compare this with the standard form of foci for a hyperbola with its transverse axis along the x-axis (or parallel to it), which is $(h \pm ae, k)$.

• By comparing the y-coordinates, we find $k = 0$. This confirms that the transverse axis lies on the x-axis.
• By comparing the x-coordinates, we have $h \pm ae = 1 \pm \sqrt{2}$.
  • This directly tells us the center of the hyperbola is $(h, k) = (1, 0)$.
  • And the distance from the center to each focus is $ae = \sqrt{2}$.

Explanation: We take this step because the foci provide crucial information about the hyperbola's position and one of its fundamental parameters, $ae$. Identifying the center $(h,k)$ is the first step in understanding the hyperbola's orientation and position.

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Step 2: Calculate the length of the semi-transverse axis, $a$

We have two pieces of information from Step 1 and the problem statement:

1. $ae = \sqrt{2}$
2. $e = \sqrt{2}$ (given eccentricity)

Now, we can substitute the value of $e$ into the equation for $ae$: $a(\sqrt{2}) = \sqrt{2}$

Dividing both sides by $\sqrt{2}$: $a = 1$

Explanation: The semi-transverse axis $a$ is a fundamental parameter of the hyperbola. We need its value, along with $b$, to calculate the length of the latus rectum. Since we know $ae$ and $e$, solving for $a$ is straightforward.

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Step 3: Calculate the square of the length of the semi-conjugate axis, $b^2$

We use the relationship between $a, b,$ and $e$ for a hyperbola: $b^2 = a^2(e^2 - 1)$

Now, substitute the values we found for $a$ and $e$:

• $a = 1$
• $e = \sqrt{2}$

$b^2 = (1)^2 ((\sqrt{2})^2 - 1)$ $b^2 = 1 (2 - 1)$ $b^2 = 1 (1)$ $b^2 = 1$

Explanation: The length of the latus rectum formula requires $b^2$. This relationship is a defining property of hyperbolas, connecting its axes lengths and eccentricity. It's crucial to use $e^2 - 1$ for a hyperbola, as opposed to $1 - e^2$ for an ellipse.

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Step 4: Calculate the Length of the Latus Rectum

Finally, we use the formula for the length of the latus rectum: $L = \frac{2b^2}{a}$

Substitute the values of $a$ and $b^2$ we have calculated:

• $a = 1$
• $b^2 = 1$

$L = \frac{2(1)}{1}$ $L = 2$

Explanation: This is the final step, directly answering the question. All previous steps were geared towards finding the necessary parameters ($a$ and $b^2$) to apply this formula.

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Tips and Common Mistakes to Avoid

1. Hyperbola vs. Ellipse: Always be careful to use the correct relationship for $b^2$. For a hyperbola, it's $b^2 = a^2(e^2 - 1)$, while for an ellipse, it's $b^2 = a^2(1 - e^2)$. A common mistake is to mix these up.
2. Orientation of Transverse Axis: The form of the foci $(h \pm ae, k)$ indicates a horizontal transverse axis. If the foci were $(h, k \pm ae)$, it would indicate a vertical transverse axis. While this doesn't change the formula for latus rectum length, it's important for writing the hyperbola's equation.
3. Careful with Square Roots: Pay attention to squaring terms like $\sqrt{2}$. $(\sqrt{2})^2 = 2$.
4. Parameter Identification: Clearly identify $h, k, ae,$ and $e$ from the given information before proceeding with calculations.

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Summary and Key Takeaway

This problem demonstrates a standard approach to working with hyperbolas:

1. Extract the hyperbola's center and the product $ae$ from the given foci.
2. Use the given eccentricity $e$ to find the semi-transverse axis $a$.
3. Utilize the fundamental relationship $b^2 = a^2(e^2 - 1)$ to find the square of the semi-conjugate axis $b^2$.
4. Finally, apply the formula for the length of the latus rectum, $L = \frac{2b^2}{a}$.

Understanding the standard forms and relationships between parameters is crucial for solving problems involving conic sections.

The final answer is $\boxed{2}$.