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

Answer

To solve the problem of finding the ratio in which the point P(-4, y) divides the line segment joining the points A(-6, 10) and B(3, -8), we can use the section formula. Let's go through the solution step by step.

### Step 1: Understand the Section Formula
The section formula states that if a point P(x, y) divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then:
- The x-coordinate of P is given by:
\[ x = \frac{mx_2 + nx_1}{m + n} \]
- The y-coordinate of P is given by:
\[ y = \frac{my_2 + ny_1}{m + n} \]

### Step 2: Assign Values
In our case:
- A(-6, 10) corresponds to (x1, y1)
- B(3, -8) corresponds to (x2, y2)
- P(-4, y) corresponds to (x, y)
- Let the ratio in which P divides AB be k:1.

### Step 3: Set Up the Equation for x-coordinate
Using the section formula for the x-coordinate:
\[ -4 = \frac{k \cdot 3 + 1 \cdot (-6)}{k + 1} \]
This simplifies to:
\[ -4(k + 1) = 3k - 6 \]
Expanding gives:
\[ -4k - 4 = 3k - 6 \]

### Step 4: Solve for k
Rearranging the equation:
\[ -4 + 6 = 3k + 4k \]
\[ 2 = 7k \]
Thus:
\[ k = \frac{2}{7} \]
This means point P divides the line segment in the ratio 2:7.

### Step 5: Set Up the Equation for y-coordinate
Now, we will find the value of y using the section formula for the y-coordinate:
\[ y = \frac{2 \cdot (-8) + 7 \cdot 10}{2 + 7} \]
Calculating this gives:
\[ y = \frac{-16 + 70}{9} = \frac{54}{9} = 6 \]

### Final Answer
The point P(-4, 6) divides the line segment joining A(-6, 10) and B(3, -8) in the ratio 2:7, and the value of y is 6.

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