Find the ratio in which the line joining `(3, 4) and (-4, 7)` is divided by y - axis. Also find the coordinates of the point os intersection.

Answer

To solve the problem of finding the ratio in which the line joining the points (3, 4) and (-4, 7) is divided by the y-axis, and to find the coordinates of the point of intersection, 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 A(3, 4) and B(-4, 7).

### Step 2: 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 m1:m2, then the coordinates of point P are given by:
\[ P\left(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\right) \]

### Step 3: Set Up the Equation for x-coordinate
Since the y-axis is where x = 0, we can set up the equation:
\[ 0 = \frac{m_1(-4) + m_2(3)}{m_1 + m_2} \]
Let’s assume the ratio is m1:m2 = λ:1. Thus, we can rewrite the equation as:
\[ 0 = \frac{\lambda(-4) + 1(3)}{\lambda + 1} \]

### Step 4: Solve for λ
From the equation:
\[ \lambda(-4) + 3 = 0 \]
This simplifies to:
\[ -4\lambda + 3 = 0 \implies 4\lambda = 3 \implies \lambda = \frac{3}{4} \]
Thus, the ratio m1:m2 is:
\[ \frac{3}{4}:1 \implies 3:4 \]

### Step 5: Find the y-coordinate of the Point of Intersection
Now, we will find the y-coordinate using the section formula:
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]
Substituting the values:
\[ y = \frac{3(7) + 4(4)}{3 + 4} \]
Calculating the numerator:
\[ 3(7) = 21 \quad \text{and} \quad 4(4) = 16 \quad \Rightarrow \quad 21 + 16 = 37 \]
Now, divide by the sum of the ratios:
\[ y = \frac{37}{7} \]

### Step 6: Final Coordinates of Intersection
The coordinates of the point of intersection on the y-axis are:
\[ (0, \frac{37}{7}) \]

### Summary of the Solution
The ratio in which the line joining (3, 4) and (-4, 7) is divided by the y-axis is **3:4**, and the coordinates of the point of intersection are **(0, 37/7)**.