In what ratio does the Y-axis divide the join of `(-4,2)` and `(8,3)`?

Answer

To find the ratio in which the Y-axis divides the line segment joining the points A(-4, 2) and B(8, 3), we can follow these steps:

### Step 1: Understand the problem
We need to find the point P on the Y-axis that divides the line segment AB in some ratio. Since P lies on the Y-axis, its x-coordinate will be 0. We can denote the coordinates of point P as (0, y).

### Step 2: Use the section formula
The section formula states that if a point P divides the line segment joining two points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of point P are given by:

\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

In our case, we can assume the ratio in which the Y-axis divides the segment AB is λ:1. Thus, we can set m = λ and n = 1.

### Step 3: Set up the equation for the x-coordinate
Since point P lies on the Y-axis, its x-coordinate is 0. Therefore, we set up the equation for the x-coordinate:

\[ 0 = \frac{λ \cdot 8 + 1 \cdot (-4)}{λ + 1} \]

### Step 4: Solve for λ
To eliminate the denominator, we multiply both sides by (λ + 1):

\[ 0 = λ \cdot 8 - 4 \]

Rearranging gives:

\[ 8λ = 4 \]

Dividing both sides by 8:

\[ λ = \frac{4}{8} = \frac{1}{2} \]

### Step 5: Write the ratio
Since we assumed the ratio was λ:1, we can now express this ratio:

\[ \text{Ratio} = \frac{1}{2}:1 = 1:2 \]

### Final Answer
The Y-axis divides the line segment joining the points (-4, 2) and (8, 3) in the ratio 1:2.

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