Calcuate the ratio in which the line joining the points (4,6) and (-5,4) is divided by the line y = 3. Also, find the co-ordinates of the point of intersection.

Answer

To solve the problem, we need to find the ratio in which the line segment joining the points \( A(4, 6) \) and \( B(-5, 4) \) is divided by the line \( y = 3 \). We will also find the coordinates of the point of intersection.

### Step 1: Identify the coordinates of points A and B
The points are given as:
- \( A(4, 6) \)
- \( B(-5, 4) \)

### Step 2: Set up the equation for the y-coordinate
The line \( y = 3 \) intersects the line segment joining points A and B. We can use the section formula to find the ratio in which this line divides the segment.

Let the ratio in which the line divides the segment be \( k:1 \). According to the section formula, the y-coordinate of the point of division (let's call it \( C \)) can be expressed as:
\[ y_C = \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1} \]
Substituting the values:
\[ 3 = \frac{k \cdot 4 + 1 \cdot 6}{k + 1} \]

### Step 3: Solve for k
Now, we will solve the equation for \( k \):
\[ 3(k + 1) = k \cdot 4 + 6 \]
Expanding both sides:
\[ 3k + 3 = 4k + 6 \]
Rearranging gives:
\[ 3 = k + 6 \]
\[ k = -3 \]

### Step 4: Interpret the ratio
The ratio \( k:1 \) is \( -3:1 \). Since the ratio is negative, it indicates that the line \( y = 3 \) divides the segment externally in the ratio \( 3:1 \).

### Step 5: Find the x-coordinate using the external division formula
Using the external division formula, the x-coordinate of point \( C \) can be calculated as:
\[ x_C = \frac{k \cdot x_2 - 1 \cdot x_1}{k - 1} \]
Substituting the values:
\[ x_C = \frac{3 \cdot (-5) - 1 \cdot 4}{3 - 1} \]
Calculating the numerator:
\[ x_C = \frac{-15 - 4}{2} = \frac{-19}{2} \]

### Step 6: Combine the coordinates
Thus, the coordinates of point \( C \) where the line \( y = 3 \) intersects the line segment \( AB \) are:
\[ C\left(-\frac{19}{2}, 3\right) \]

### Final Answer
The ratio in which the line joining the points \( (4, 6) \) and \( (-5, 4) \) is divided by the line \( y = 3 \) is \( 3:1 \) externally, and the coordinates of the point of intersection are \( C\left(-\frac{19}{2}, 3\right) \).

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