Find the ratio in which the point (2, y) divides the line segment(4,3) and (6,3). hence find the value of y

Answer

To solve the problem of finding the ratio in which the point (2, y) divides the line segment joining the points (4, 3) and (6, 3), we can follow these steps:

### Step 1: Identify the points and the coordinates
We have the points:
- Point A = (4, 3)
- Point B = (6, 3)
- Point P = (2, y) which divides the line segment AB.

### Step 2: Set up the ratio
Let the point P divide the line segment AB in the ratio \( m: n \). We can assume \( m = 1 \) and \( n = \lambda \). Thus, the ratio is \( 1: \lambda \).

### Step 3: Use the section formula for the x-coordinates
According to the section formula, the x-coordinate of point P can be expressed as:
\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \]
Substituting the known values:
\[ 2 = \frac{1 \cdot 6 + \lambda \cdot 4}{1 + \lambda} \]

### Step 4: Solve for λ
Multiplying both sides by \( (1 + \lambda) \):
\[ 2(1 + \lambda) = 6 + 4\lambda \]
Expanding and rearranging gives:
\[ 2 + 2\lambda = 6 + 4\lambda \]
\[ 2 - 6 = 4\lambda - 2\lambda \]
\[ -4 = 2\lambda \]
\[ \lambda = -2 \]

### Step 5: Find the ratio
The ratio in which point P divides the line segment AB is:
\[ 1 : \lambda = 1 : -2 \]
This can be interpreted as the ratio \( -1 : 2 \) (indicating that the point P lies outside the segment).

### Step 6: Use the section formula for the y-coordinates
Now, we can find the value of y using the section formula for the y-coordinates:
\[ y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \]
Substituting the known values:
\[ y = \frac{1 \cdot 3 + (-2) \cdot 3}{1 + (-2)} \]
\[ y = \frac{3 - 6}{1 - 2} \]
\[ y = \frac{-3}{-1} = 3 \]

### Final Answer
Thus, the value of \( y \) is \( 3 \).

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