Find the ratio in which the segment joining the points `A(-6, 10) and B(3, -8)` is divided by the point `(-4, 6)`.

Answer

To find the ratio in which the segment joining the points A(-6, 10) and B(3, -8) is divided by the point C(-4, 6), we can use the section formula. The section formula states that if a point divides the line segment joining two points in the ratio m:n, then the coordinates of the point can be calculated as follows:

\[ C\left(x, y\right) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

Where:
- \(A(x_1, y_1) = (-6, 10)\)
- \(B(x_2, y_2) = (3, -8)\)
- \(C(x, y) = (-4, 6)\)

Let’s assume the ratio in which point C divides the segment AB is \(m:n\). We can denote \(m\) as \(λ\) and \(n\) as \(1\) (for simplicity).

### Step 1: Set up the equations using the section formula

Using the section formula for the x-coordinates:

\[ -4 = \frac{λ \cdot 3 + 1 \cdot (-6)}{λ + 1} \]

Using the section formula for the y-coordinates:

\[ 6 = \frac{λ \cdot (-8) + 1 \cdot 10}{λ + 1} \]

### Step 2: Solve the first equation for \(λ\)

From the x-coordinate equation:

\[ -4(λ + 1) = 3λ - 6 \]

Expanding and rearranging gives:

\[ -4λ - 4 = 3λ - 6 \]

Combining like terms:

\[ -4λ - 3λ = -6 + 4 \]

\[ -7λ = -2 \]

\[ λ = \frac{2}{7} \]

### Step 3: Solve the second equation for verification

Now, substituting \(λ = \frac{2}{7}\) into the y-coordinate equation:

\[ 6 = \frac{\frac{2}{7} \cdot (-8) + 10}{\frac{2}{7} + 1} \]

Calculating the denominator:

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

Now substituting into the equation:

\[ 6 = \frac{-\frac{16}{7} + 10}{\frac{9}{7}} \]

Calculating the numerator:

\[ 10 = \frac{70}{7} \Rightarrow -\frac{16}{7} + \frac{70}{7} = \frac{54}{7} \]

So:

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

This confirms our value of \(λ\).

### Step 4: Find the ratio

The ratio \(m:n\) is \(λ:1\), which is \(\frac{2}{7}:1\). Therefore, the ratio in which the segment joining points A and B is divided by point C is:

\[ 2:7 \]

### Final Answer

The ratio in which the segment joining the points A(-6, 10) and B(3, -8) is divided by the point C(-4, 6) is \(2:7\).

---