Find the ratio in which the y-axis cuts the line joining the points (4, 5) and (-10, 2)

Options

A. `2 : 5`
B. ` 5 : 2 `
C. ` 3 : 2 `
D. None of these

Answer

To find the ratio in which the y-axis cuts the line joining the points (4, 5) and (-10, 2), we can follow these steps:

### Step 1: Identify the Points
We have two points:
- Point A: (4, 5)
- Point B: (-10, 2)

### Step 2: Set Up the Section Formula
The y-axis is represented by the line where x = 0. We need to find the point where the line joining points A and B intersects the y-axis. Let's denote this intersection point as P(0, y).

According to the section formula, 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 are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

### Step 3: Assign the Ratio
Let's assume the ratio in which the y-axis cuts the line is n:1. This means:
- m = n (the part towards point B)
- n = 1 (the part towards point A)

### Step 4: Substitute the Values
Using the section formula, we can express the coordinates of point P as:
\[ P\left(0, \frac{n \cdot 2 + 1 \cdot 5}{n + 1}\right) \]
For the x-coordinate to be 0, we need to find the value of n such that:
\[ \frac{-10n + 4}{n + 1} = 0 \]

### Step 5: Solve for n
Setting the numerator equal to zero:
\[ -10n + 4 = 0 \]
\[ 10n = 4 \]
\[ n = \frac{4}{10} = \frac{2}{5} \]

### Step 6: Determine the Ratio
Since we assumed the ratio is n:1, we have:
- The ratio is \( \frac{2}{5} : 1 \)
- This can be expressed as \( 2 : 5 \).

### Final Answer
Thus, the ratio in which the y-axis cuts the line joining the points (4, 5) and (-10, 2) is **2:5**.

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