Find the ratio in which the line x - 3y = 0 divides the line segment joining the points (-2,-5) and (6,3). Find the coordinates of the point of intersection.

Answer

To solve the problem step by step, we will find the ratio in which the line \(x - 3y = 0\) divides the line segment joining the points \((-2, -5)\) and \((6, 3)\), and also find the coordinates of the point of intersection.

### Step 1: Identify Points and Set Up the Section Formula
Let the points be:
- \(A(-2, -5)\) (Point 1)
- \(B(6, 3)\) (Point 2)

Let the point \(P\) divide the line segment \(AB\) in the ratio \(k:1\).

### Step 2: Use the Section Formula
The coordinates of point \(P\) that divides the line segment \(AB\) in the ratio \(k:1\) are given by the section formula:
\[ P\left(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\right) \]
Where:
- \(m_1 = k\)
- \(m_2 = 1\)
- \(x_1 = -2\), \(y_1 = -5\)
- \(x_2 = 6\), \(y_2 = 3\)

Thus, the coordinates of point \(P\) are:
\[ P\left(\frac{k \cdot 6 + 1 \cdot (-2)}{k + 1}, \frac{k \cdot 3 + 1 \cdot (-5)}{k + 1}\right) \]

### Step 3: Substitute into the Line Equation
Since point \(P\) lies on the line \(x - 3y = 0\), we can substitute the coordinates of \(P\) into this equation:
\[ \frac{6k - 2}{k + 1} - 3\left(\frac{3k - 5}{k + 1}\right) = 0 \]

### Step 4: Simplify the Equation
Multiply through by \(k + 1\) to eliminate the denominator:
\[ 6k - 2 - 3(3k - 5) = 0 \]
Expanding gives:
\[ 6k - 2 - 9k + 15 = 0 \]
Combining like terms:
\[ -3k + 13 = 0 \]

### Step 5: Solve for \(k\)
Rearranging gives:
\[ 3k = 13 \implies k = \frac{13}{3} \]

### Step 6: Find the Ratio
The ratio in which the line divides the segment \(AB\) is \(k:1\), which is:
\[ \frac{13}{3}:1 \text{ or } 13:3 \]

### Step 7: Find the Coordinates of Point \(P\)
Now, substitute \(k = \frac{13}{3}\) back into the coordinates of \(P\):
\[ P\left(\frac{\frac{13}{3} \cdot 6 - 2}{\frac{13}{3} + 1}, \frac{\frac{13}{3} \cdot 3 - 5}{\frac{13}{3} + 1}\right) \]

Calculating the x-coordinate:
\[ x = \frac{\frac{78}{3} - 2}{\frac{16}{3}} = \frac{26 - 2}{\frac{16}{3}} = \frac{24}{\frac{16}{3}} = \frac{24 \cdot 3}{16} = \frac{72}{16} = \frac{9}{2} \]

Calculating the y-coordinate:
\[ y = \frac{\frac{39}{3} - 5}{\frac{16}{3}} = \frac{13 - 5}{\frac{16}{3}} = \frac{8}{\frac{16}{3}} = \frac{8 \cdot 3}{16} = \frac{24}{16} = \frac{3}{2} \]

### Final Result
The coordinates of point \(P\) are:
\[ P\left(\frac{9}{2}, \frac{3}{2}\right) \]
The ratio in which the line divides the segment is \(13:3\).