In what ratio is the line segment joining the points (-3,2) and (6,1) is divided by Y-axis?

Options

A. `1:3`
B. `2:1`
C. `1:2`
D. `3:1`

Answer

To find the ratio in which the line segment joining the points (-3, 2) and (6, 1) is divided by the Y-axis, we can follow these steps:

### Step 1: Identify the points
Let the two points be:
- Point A = (-3, 2)
- Point B = (6, 1)

### Step 2: Understand the Y-axis
The Y-axis is defined by the equation x = 0. We need to find the point on the Y-axis where the line segment AB intersects.

### Step 3: Use the section formula
The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of point P can be given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
In our case, we want to find the point P on the Y-axis, which means the x-coordinate of P will be 0.

### Step 4: Set up the equation for the x-coordinate
Let the ratio in which the Y-axis divides the segment be λ:1. Then, we can substitute into the section formula:
\[ 0 = \frac{λ \cdot 6 + 1 \cdot (-3)}{λ + 1} \]

### Step 5: Solve for λ
Now, we can set up the equation:
\[ 0 = \frac{6λ - 3}{λ + 1} \]
This implies:
\[ 6λ - 3 = 0 \]
Solving for λ gives:
\[ 6λ = 3 \implies λ = \frac{3}{6} = \frac{1}{2} \]

### Step 6: Write the ratio
The ratio in which the Y-axis divides the line segment is λ:1, which is:
\[ \frac{1}{2}:1 \text{ or } 1:2 \]

### Final Answer
The line segment joining the points (-3, 2) and (6, 1) is divided by the Y-axis in the ratio **1:2**.

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