https://r2.erweima.ai/mathbot/image/17192138908915jzzlone.jpg

Let's start by analyzing the content of the image. 

The questions are in Chinese and involve mathematical problems related to coordinates and distances in a plane. Here's a breakdown and translation of the given problems:

1. The definition of "friend distance" (友好距离) between two points \( P_1 (x_1, y_1) \) and \( P_2 (x_2, y_2) \) on a coordinate plane:
   \[
   \text{Friend Distance} = |x_1 - x_2| + |y_1 - y_2|
   \]

2. Given point \( A(-3, 0) \), determine:
   - (1) The coordinates of point \( B \) that minimizes the friend distance between \( A \) and \( B \).
   - (2) The coordinates of point \( C(m, 2) \) that minimizes the friend distance under the condition \( m < 0 \).

Here are the detailed steps for solving the given problems:

### Problem (1)

For minimizing the friend distance \( |x_1 - x_2| + |y_1 - y_2| \):

- Given \( A(-3, 0) \), the coordinates of point \( B \) should be \( B(2, 0) \) because \( y \) coordinates are the same and the minimum value for \( |x_1 - x_2| \) is obtained when the points are on the same vertical line.

### Problem (2)

For minimizing the friend distance between \( C(m, 2) \) and \( B \):

- Given \( C(m, 2) \) and \( m < 0 \), the coordinates of point \( C \) that minimize the friend distance with condition \( m < 0 \) can be found by minimizing \( |x_1 - x_2| \).

- The distance function to minimize is \( \text{Friend Distance} = |m - 2| + |2 - y_2| \).

Let's calculate for the specific cases:

1. For \( y_2 = 2 \) (since the minimum y-distance is when y-coordinates are equal):
   \[
   \text{Friend Distance} = |m - 2| + |2 - 2| = |m - 2|
   \]

### Solution Summary:
- For point \( B \), the coordinates minimizing the friend distance with point \( A(-3, 0) \) are \( B(2, 0) \).
- For point \( C(m, 2) \), where \( m < 0 \), the coordinates minimizing the friend distance with \( B \) are directly influenced by minimizing the \( |m - 2| \) term.

If you need more detailed calculations or have any specific queries, let me know! Here are some questions you can ask next:

1. Can you explain how to derive the friend distance formula?
2. How can the concept of friend distance be applied to other geometric problems?
3. What are some properties of the friend distance in different coordinate systems?
4. Can we find a general formula for minimizing friend distance between any two points?
5. How does the condition \( m < 0 \) specifically impact the solution?

**Tip:** Remember that understanding the underlying principles of geometric distances can significantly enhance your problem-solving skills in coordinate geometry.

Explore how to minimize friend distance between points in a coordinate plane using the distance formula. Detailed solutions for minimizing distances between points A(-3, 0), B(2, 0), and C(m, 2) under the condition m < 0. Ideal for high school students learning coordinate geometry.

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