If `P ( 9a -2, -b)` divides the line segment joining the points `A (31 +1, -3) and B ( 81, 5)` in the ratio `3:1:` Find the values of a and b.

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given that point \( P(9a - 2, -b) \) divides the line segment joining points \( A(32, -3) \) and \( B(81, 5) \) in the ratio \( 3:1 \).

### Step 1: Identify the coordinates of points A and B
The coordinates of point \( A \) are given as \( (31 + 1, -3) \), which simplifies to:
\[ A(32, -3) \]
The coordinates of point \( B \) are given as \( (81, 5) \).

### Step 2: Apply the section formula
The section formula states that if a point \( P(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, \( m = 3 \) and \( n = 1 \).

### Step 3: Calculate the x-coordinate of P
Substituting the values into the formula for the x-coordinate:
\[ x_P = \frac{3 \cdot 81 + 1 \cdot 32}{3 + 1} = \frac{243 + 32}{4} = \frac{275}{4} \]

### Step 4: Calculate the y-coordinate of P
Now, substituting the values into the formula for the y-coordinate:
\[ y_P = \frac{3 \cdot 5 + 1 \cdot (-3)}{3 + 1} = \frac{15 - 3}{4} = \frac{12}{4} = 3 \]

### Step 5: Set up equations for a and b
We know that the coordinates of point \( P \) are given as \( (9a - 2, -b) \). Therefore, we can set up the following equations:
1. \( 9a - 2 = \frac{275}{4} \)
2. \( -b = 3 \) or \( b = -3 \)

### Step 6: Solve for b
From the second equation:
\[ b = -3 \]

### Step 7: Solve for a
Now, we will solve the first equation for \( a \):
\[ 9a - 2 = \frac{275}{4} \]
Adding 2 to both sides:
\[ 9a = \frac{275}{4} + 2 = \frac{275}{4} + \frac{8}{4} = \frac{283}{4} \]
Now, divide both sides by 9:
\[ a = \frac{283}{36} \]

### Final Values
Thus, the values of \( a \) and \( b \) are:
\[ a = \frac{283}{36}, \quad b = -3 \]