In what ratio y-axis divides the line segment joining the points (3,4) and (–2, 1)?

Answer

To find the ratio in which the y-axis divides the line segment joining the points \( A(3, 4) \) and \( B(-2, 1) \), we can follow these steps:

### Step 1: Identify the Points
Let \( A(3, 4) \) and \( B(-2, 1) \) be the two points.

### Step 2: Find the Equation of the Line
We will use the two-point form of the equation of a line, which is given by:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]
Here, \( (x_1, y_1) = (3, 4) \) and \( (x_2, y_2) = (-2, 1) \).

Substituting these values into the equation:
\[ y - 4 = \frac{1 - 4}{-2 - 3} (x - 3) \]
This simplifies to:
\[ y - 4 = \frac{-3}{-5} (x - 3) \]
\[ y - 4 = \frac{3}{5} (x - 3) \]

### Step 3: Rearranging the Equation
Now, let's rearrange this equation:
\[ y - 4 = \frac{3}{5}x - \frac{9}{5} \]
Adding 4 to both sides:
\[ y = \frac{3}{5}x + 4 - \frac{9}{5} \]
Converting 4 to a fraction:
\[ y = \frac{3}{5}x + \frac{20}{5} - \frac{9}{5} \]
\[ y = \frac{3}{5}x + \frac{11}{5} \]

### Step 4: Find the Intersection with the Y-axis
To find the point where the line intersects the y-axis, we set \( x = 0 \):
\[ y = \frac{3}{5}(0) + \frac{11}{5} = \frac{11}{5} \]
So, the coordinates of point \( C \) where the line intersects the y-axis are \( (0, \frac{11}{5}) \).

### Step 5: Use the Section Formula
Let the y-axis divide the segment \( AB \) in the ratio \( m:n \). According to the section formula, the coordinates of point \( C \) can be expressed as:
\[ C\left(0, \frac{11}{5}\right) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Substituting \( A(3, 4) \) and \( B(-2, 1) \):
\[ 0 = \frac{m(-2) + n(3)}{m+n} \]
This gives us:
\[ 0 = -2m + 3n \quad \text{(1)} \]

For the y-coordinates:
\[ \frac{11}{5} = \frac{m(1) + n(4)}{m+n} \]
Multiplying both sides by \( m+n \):
\[ 11(m+n) = 5(m + 4n) \]
Expanding gives:
\[ 11m + 11n = 5m + 20n \]
Rearranging gives:
\[ 6m - 9n = 0 \quad \text{(2)} \]

### Step 6: Solve the Equations
From equation (1):
\[ 2m = 3n \implies \frac{m}{n} = \frac{3}{2} \]
Let \( m = 3k \) and \( n = 2k \) for some \( k \).

### Step 7: Conclusion
Thus, the ratio in which the y-axis divides the line segment joining points \( A \) and \( B \) is:
\[ m:n = 3:2 \]