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 \((-4, 2)\) and \((8, 3)\), we can follow these steps:

### Step 1: Understand the problem
We need to find the ratio in which the Y-axis (where \(x = 0\)) divides the line segment joining the points \((-4, 2)\) and \((8, 3)\).

### Step 2: Set up the ratio
Let the ratio in which the Y-axis divides the segment be \(k:1\). This means that the point of division (let's call it \(P\)) can be expressed in terms of \(k\) and the coordinates of the two points.

### Step 3: Use the section formula
The coordinates of point \(P\) dividing the line segment joining points \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m:n\) can be found using the section formula:
\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In our case, \(m = k\) and \(n = 1\), so:
\[ P\left( \frac{k \cdot 8 + 1 \cdot (-4)}{k + 1}, \frac{k \cdot 3 + 1 \cdot 2}{k + 1} \right) \]

### Step 4: Set the x-coordinate to 0
Since point \(P\) lies on the Y-axis, its x-coordinate must be 0:
\[ \frac{8k - 4}{k + 1} = 0 \]

### Step 5: Solve for \(k\)
Setting the numerator equal to zero gives:
\[ 8k - 4 = 0 \]
\[ 8k = 4 \]
\[ k = \frac{4}{8} = \frac{1}{2} \]

### Step 6: Write the ratio
The ratio in which the Y-axis divides the segment is \(k:1 = \frac{1}{2}:1\). To express this in whole numbers, we can multiply both parts of the ratio by 2:
\[ 1:2 \]

### 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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