Calculate the ratio in which the line segment A(6, 5) and B(4, -3) is divided by the line y = 2.

Answer

To calculate the ratio in which the line segment AB, with points A(6, 5) and B(4, -3), is divided by the line y = 2, we can follow these steps:

### Step 1: Identify the coordinates of points A and B
- Point A has coordinates (6, 5).
- Point B has coordinates (4, -3).

### Step 2: Understand the line y = 2
- The line y = 2 is a horizontal line, meaning it runs parallel to the x-axis. Any point on this line will have a y-coordinate of 2.

### Step 3: Use the section formula
- Let the point P(x, 2) divide the line segment AB in the ratio k:1. According to the section formula, the y-coordinate of point P can be expressed as:
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]
where \( m_1 = k \) and \( m_2 = 1 \).

### Step 4: Substitute the known values into the formula
- Here, \( y_1 = 5 \) (from point A) and \( y_2 = -3 \) (from point B). Thus, we have:
\[ 2 = \frac{k \cdot (-3) + 1 \cdot 5}{k + 1} \]

### Step 5: Clear the fraction by multiplying both sides by (k + 1)
- This gives us:
\[ 2(k + 1) = -3k + 5 \]
Expanding the left side:
\[ 2k + 2 = -3k + 5 \]

### Step 6: Rearrange the equation
- Move all terms involving k to one side and constant terms to the other:
\[ 2k + 3k = 5 - 2 \]
This simplifies to:
\[ 5k = 3 \]

### Step 7: Solve for k
- Divide both sides by 5:
\[ k = \frac{3}{5} \]

### Step 8: Write the ratio
- The ratio in which the line segment AB is divided by the line y = 2 is:
\[ AP:PB = k:1 = \frac{3}{5}:1 = 3:5 \]

### Final Answer
The required ratio is **3:5**.

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