In what ratio does the point P(-4, y) divide the line segment joining the point A(-6, 10) and B(3, -8) if it lies on AB. Also, find the value of y.

Answer

To solve the problem, we need to determine the ratio in which the point P(-4, y) divides the line segment joining points A(-6, 10) and B(3, -8), and also find the value of y.

### Step 1: 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 can be expressed as:
\[ P\left(x, y\right) = \left(\frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\right) \]
Here, we have:
- \(A(-6, 10)\) → \(x_1 = -6, y_1 = 10\)
- \(B(3, -8)\) → \(x_2 = 3, y_2 = -8\)
- \(P(-4, y)\) → \(x = -4, y = y\)

### Step 2: Set Up the Equation for x-coordinate
Using the x-coordinate of point P:
\[ -4 = \frac{m_1 \cdot 3 + m_2 \cdot (-6)}{m_1 + m_2} \]

### Step 3: Cross Multiply
Cross-multiplying gives:
\[ -4(m_1 + m_2) = 3m_1 - 6m_2 \]
Expanding this, we have:
\[ -4m_1 - 4m_2 = 3m_1 - 6m_2 \]

### Step 4: Rearranging the Equation
Rearranging the terms:
\[ -4m_1 - 3m_1 = -6m_2 + 4m_2 \]
This simplifies to:
\[ -7m_1 = -2m_2 \]
Dividing both sides by -1:
\[ 7m_1 = 2m_2 \]
Thus, we can write:
\[ \frac{m_1}{m_2} = \frac{2}{7} \]

### Step 5: Conclusion for the Ratio
The ratio in which point P divides the line segment AB is:
\[ m_1:m_2 = 2:7 \]

### Step 6: Set Up the Equation for y-coordinate
Now, we will find the value of y using the y-coordinate:
\[ y = \frac{m_1 \cdot (-8) + m_2 \cdot 10}{m_1 + m_2} \]
Substituting \(m_1 = 2\) and \(m_2 = 7\):
\[ y = \frac{2 \cdot (-8) + 7 \cdot 10}{2 + 7} \]

### Step 7: Calculate y
Calculating the numerator:
\[ y = \frac{-16 + 70}{9} = \frac{54}{9} = 6 \]

### Final Answer
The point P divides the line segment joining A and B in the ratio \(2:7\) and the value of \(y\) is \(6\).