If the point P(-1,2) divides the join of points A(2, 5) and B(a, b) in the ratio 3 : 4, find the value of `a xx b - a`.

Answer

To solve the problem step-by-step, we will use the section formula to find the coordinates of point B (a, b) given that point P(-1, 2) divides the line segment joining points A(2, 5) and B(a, b) in the ratio 3:4.

### Step 1: Understand 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 (x, y) can be calculated as follows:
- \( x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \)
- \( y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \)

### Step 2: Assign Values
Here, we have:
- Point A = (2, 5) → \( x_1 = 2, y_1 = 5 \)
- Point B = (a, b) → \( x_2 = a, y_2 = b \)
- Point P = (-1, 2) → \( x = -1, y = 2 \)
- The ratio m:n = 3:4 → \( m = 3, n = 4 \)

### Step 3: Set Up the Equations
Using the section formula for the x-coordinate:
\[ -1 = \frac{3a + 4 \cdot 2}{3 + 4} \]
This simplifies to:
\[ -1 = \frac{3a + 8}{7} \]

Using the section formula for the y-coordinate:
\[ 2 = \frac{3b + 4 \cdot 5}{3 + 4} \]
This simplifies to:
\[ 2 = \frac{3b + 20}{7} \]

### Step 4: Solve for a
From the x-coordinate equation:
\[ -1 \cdot 7 = 3a + 8 \]
\[ -7 = 3a + 8 \]
\[ 3a = -7 - 8 \]
\[ 3a = -15 \]
\[ a = -5 \]

### Step 5: Solve for b
From the y-coordinate equation:
\[ 2 \cdot 7 = 3b + 20 \]
\[ 14 = 3b + 20 \]
\[ 3b = 14 - 20 \]
\[ 3b = -6 \]
\[ b = -2 \]

### Step 6: Calculate \( a \cdot b - a \)
Now that we have \( a = -5 \) and \( b = -2 \):
\[ a \cdot b - a = (-5) \cdot (-2) - (-5) \]
\[ = 10 + 5 \]
\[ = 15 \]

### Final Answer
The value of \( a \cdot b - a \) is **15**.

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