In what ratio does the point P(-4, y) divide the line segment joining the points A(-6, 10) and B(3, -8)? Find the value of y.

Answer

To find the ratio in which the point P(-4, y) divides the line segment joining the points A(-6, 10) and B(3, -8), we will 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 m:n, then the coordinates of point P can be given by:

\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

### Step 1: Set up the equation for the x-coordinate

Let the ratio in which point P divides the line segment AB be λ:1. Therefore, we can write:

\[ P_x = \frac{3\lambda + (-6)}{\lambda + 1} \]

Given that \(P_x = -4\), we can set up the equation:

\[ -4 = \frac{3\lambda - 6}{\lambda + 1} \]

### Step 2: Cross-multiply to eliminate the fraction

Cross-multiplying gives us:

\[ -4(\lambda + 1) = 3\lambda - 6 \]

Expanding this, we get:

\[ -4\lambda - 4 = 3\lambda - 6 \]

### Step 3: Rearrange the equation to solve for λ

Now, we will rearrange the equation:

\[ -4\lambda - 3\lambda = -6 + 4 \]

This simplifies to:

\[ -7\lambda = -2 \]

Dividing both sides by -7 gives:

\[ \lambda = \frac{2}{7} \]

### Step 4: Find the ratio

The ratio in which point P divides the line segment AB is:

\[ \frac{2}{7} : 1 \quad \text{or} \quad 2 : 7 \]

### Step 5: Set up the equation for the y-coordinate

Now, we will find the value of y using the y-coordinates of points A and B:

\[ P_y = \frac{(-8)\lambda + 10}{\lambda + 1} \]

Substituting \(\lambda = \frac{2}{7}\):

\[ y = \frac{-8 \left(\frac{2}{7}\right) + 10}{\frac{2}{7} + 1} \]

### Step 6: Simplify the equation

Calculating the numerator:

\[ -8 \left(\frac{2}{7}\right) + 10 = -\frac{16}{7} + 10 = -\frac{16}{7} + \frac{70}{7} = \frac{54}{7} \]

Calculating the denominator:

\[ \frac{2}{7} + 1 = \frac{2}{7} + \frac{7}{7} = \frac{9}{7} \]

### Step 7: Final calculation for y

Now substituting back into the equation for y:

\[ y = \frac{\frac{54}{7}}{\frac{9}{7}} = \frac{54}{9} = 6 \]

### Conclusion

Thus, the value of y is \(6\) and the ratio in which point P divides the line segment AB is \(2:7\).

---