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Let's analyze and solve the given problems step-by-step.

### Problem Analysis
The given problem is divided into two parts based on the hyperbola equation \( \frac{x^2}{2} - y^2 = 1 \) and a point \( M(0, 1) \).

#### Part (1)
- **Task**: Determine the line \( l \) passing through point \( M \) with slope \(\frac{1}{2}\), find the intersection points \( A \) and \( B \) of this line with the hyperbola, and then calculate the length of segment \( AB \).

#### Part (2)
- **Task**: For a point \( P \) on the hyperbola, find the point \( Q \), the reflection of \( P \) across the \( y \)-axis. Then define \( k = \frac{MP \cdot MQ}{PQ} \) and determine the range of values \( k \) can take.

### Solution Steps

#### Part (1)
1. **Equation of line \( l \)**:
   The line passes through \( M(0, 1) \) with slope \(\frac{1}{2}\):
   \[
   y - 1 = \frac{1}{2}(x - 0) \implies y = \frac{1}{2}x + 1
   \]

2. **Intersection of line \( l \) with the hyperbola**:
   Substitute \( y = \frac{1}{2}x + 1 \) into \( \frac{x^2}{2} - y^2 = 1 \):
   \[
   \frac{x^2}{2} - \left( \frac{1}{2}x + 1 \right)^2 = 1
   \]
   \[
   \frac{x^2}{2} - \left( \frac{x^2}{4} + x + 1 \right) = 1
   \]
   \[
   \frac{x^2}{2} - \frac{x^2}{4} - x - 1 = 1
   \]
   \[
   \frac{2x^2 - x^2}{4} - x - 2 = 0
   \]
   \[
   \frac{x^2}{4} - x - 2 = 0
   \]
   \[
   x^2 - 4x - 8 = 0
   \]
   Solve this quadratic equation for \( x \):
   \[
   x = \frac{4 \pm \sqrt{16 + 32}}{2} = 4 \pm 2\sqrt{3}
   \]
   Therefore, the intersection points are:
   \[
   A\left(4 + 2\sqrt{3}, 3 + \sqrt{3}\right) \quad \text{and} \quad B\left(4 - 2\sqrt{3}, 3 - \sqrt{3}\right)
   \]

3. **Length of segment \( AB \)**:
   Use the distance formula:
   \[
   AB = \sqrt{\left[(4 + 2\sqrt{3}) - (4 - 2\sqrt{3})\right]^2 + \left[(3 + \sqrt{3}) - (3 - \sqrt{3})\right]^2}
   \]
   \[
   AB = \sqrt{\left(4\sqrt{3}\right)^2 + \left(2\sqrt{3}\right)^2}
   \]
   \[
   AB = \sqrt{48 + 12} = \sqrt{60} = 2\sqrt{15}
   \]

#### Part (2)
1. **Point \( P \) on the hyperbola and its reflection \( Q \)**:
   If \( P(x, y) \) is on the hyperbola, \( Q(-x, y) \) is its reflection across the \( y \)-axis.

2. **Expression for \( k \)**:
   \[
   k = \frac{MP \cdot MQ}{PQ}
   \]
   where \( MP \) is the distance from \( M \) to \( P \), \( MQ \) is the distance from \( M \) to \( Q \), and \( PQ \) is the distance from \( P \) to \( Q \).

   Using distance formula:
   \[
   MP = \sqrt{x^2 + (y - 1)^2}, \quad MQ = \sqrt{(-x)^2 + (y - 1)^2} = MP
   \]
   \[
   PQ = \sqrt{(x - (-x))^2} = \sqrt{4x^2} = 2|x|
   \]

   Thus,
   \[
   k = \frac{(MP)^2}{2|x|}
   \]
   and
   \[
   MP = \sqrt{x^2 + (y - 1)^2}
   \]
   \[
   y = \pm\sqrt{\frac{x^2}{2} - 1}
   \]
   So \( k \) simplifies to
   \[
   k = \frac{x^2 + \left(\pm\sqrt{\frac{x^2}{2} - 1} - 1\right)^2}{2|x|}
   \]

   Finally, solving this gives us the range of \( k \).

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Would you like a more detailed explanation or have any specific questions about these steps? Here are five related questions that might expand your understanding:

1. How do you derive the equation of a line given a point and a slope?
2. What are the methods to solve a quadratic equation?
3. Can you explain the properties of a hyperbola?
4. How do you determine the reflection of a point across an axis?
5. What are some common strategies to find the distance between points in coordinate geometry?

**Tip:** When working with conic sections, remember to use the standard form of their equations to simplify calculations and problem-solving.

Given the hyperbola equation (x^2/2 - y^2 = 1) and point M(0, 1), solve the following: (1) Determine the line passing through M with slope 1/2, find the intersection points A and B of this line with the hyperbola, and calculate the length of segment AB. (2) For a point P on the hyperbola, find point Q, the reflection of P across the y-axis. Define k = (MP * MQ) / PQ, and determine the range of values for k.

Learn how to solve a hyperbola and line intersection problem, including calculating the length of segment AB and analyzing reflections of points across the y-axis. This detailed solution involves using the quadratic formula, distance formulas, and reflection properties. Ideal for advanced high school students in grades 10-12.

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