The point P divides the join of (2, 1) and (-3, 6) in the ratio 2 : 3. Does P lie on the line `x - 5y + 15 = 0` ?

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Identify the Points and the Ratio
We are given two points:
- Point A (x₁, y₁) = (2, 1)
- Point B (x₂, y₂) = (-3, 6)

The point P divides the line segment joining A and B in the ratio 2:3.

### Step 2: Use the Section Formula
The coordinates of point P (x, y) can be calculated using the section formula:
\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \]
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]
Where \(m_1\) and \(m_2\) are the ratios in which point P divides the segment. Here, \(m_1 = 2\) and \(m_2 = 3\).

### Step 3: Calculate the x-coordinate of P
Substituting the values into the formula for x:
\[ x = \frac{2 \cdot (-3) + 3 \cdot 2}{2 + 3} \]
Calculating this:
\[ x = \frac{-6 + 6}{5} = \frac{0}{5} = 0 \]

### Step 4: Calculate the y-coordinate of P
Now substituting the values into the formula for y:
\[ y = \frac{2 \cdot 6 + 3 \cdot 1}{2 + 3} \]
Calculating this:
\[ y = \frac{12 + 3}{5} = \frac{15}{5} = 3 \]

### Step 5: Coordinates of Point P
Thus, the coordinates of point P are:
\[ P(0, 3) \]

### Step 6: Check if Point P Lies on the Given Line
The equation of the line is:
\[ x - 5y + 15 = 0 \]
Substituting the coordinates of point P (0, 3) into the equation:
\[ 0 - 5(3) + 15 = 0 \]
Calculating this:
\[ 0 - 15 + 15 = 0 \]
This simplifies to:
\[ 0 = 0 \]

### Conclusion
Since the left-hand side (LHS) equals the right-hand side (RHS), point P lies on the line.

### Final Answer
Yes, point P lies on the line \(x - 5y + 15 = 0\).

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