Find the point which divides the line segment joining
(4,5), (-3,4) in the ratio -6 : 5

Answer

To find the point that divides the line segment joining the points (4,5) and (-3,4) in the ratio -6:5, we can 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 (x, y) can be calculated as follows:

\[ P\left(x, y\right) = \left(\frac{m1 \cdot x2 + m2 \cdot x1}{m1 + m2}, \frac{m1 \cdot y2 + m2 \cdot y1}{m1 + m2}\right) \]

### Step 1: Identify the coordinates and the ratio
Let:
- A (x1, y1) = (4, 5)
- B (x2, y2) = (-3, 4)
- m1 = -6
- m2 = 5

### Step 2: Substitute the values into the formula
Using the section formula, we substitute the values:

\[ P\left(x, y\right) = \left(\frac{-6 \cdot (-3) + 5 \cdot 4}{-6 + 5}, \frac{-6 \cdot 4 + 5 \cdot 5}{-6 + 5}\right) \]

### Step 3: Calculate the x-coordinate
Calculate the x-coordinate:

\[ x = \frac{-6 \cdot (-3) + 5 \cdot 4}{-6 + 5} = \frac{18 + 20}{-1} = \frac{38}{-1} = -38 \]

### Step 4: Calculate the y-coordinate
Calculate the y-coordinate:

\[ y = \frac{-6 \cdot 4 + 5 \cdot 5}{-6 + 5} = \frac{-24 + 25}{-1} = \frac{1}{-1} = -1 \]

### Step 5: Combine the coordinates
Thus, the point P that divides the line segment in the given ratio is:

\[ P(-38, -1) \]

### Final Answer
The point which divides the line segment joining (4, 5) and (-3, 4) in the ratio -6:5 is (-38, -1).

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