Point A and B have co-ordinates (7, -3) and (1,9) respectively. Find :
the value of 'p' if (-2, p) lies on it. 

Answer

To find the value of 'p' such that the point (-2, p) lies on the line joining points A (7, -3) and B (1, 9), we can follow these steps:

### Step 1: Identify the coordinates of points A and B
We have:
- Point A: (7, -3)
- Point B: (1, 9)

### Step 2: Use the formula for the equation of a line through two points
The equation of a line through two points (x1, y1) and (x2, y2) can be expressed as:
\[ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1) \]

Substituting the coordinates of points A and B:
- \( x_1 = 7, y_1 = -3 \)
- \( x_2 = 1, y_2 = 9 \)

### Step 3: Substitute the values into the formula
\[ y - (-3) = \frac{9 - (-3)}{1 - 7} (x - 7) \]

### Step 4: Simplify the equation
Calculating the slope:
\[ y + 3 = \frac{9 + 3}{1 - 7} (x - 7) \]
\[ y + 3 = \frac{12}{-6} (x - 7) \]
\[ y + 3 = -2 (x - 7) \]

### Step 5: Distribute and rearrange the equation
Expanding the equation:
\[ y + 3 = -2x + 14 \]
Now, rearranging gives:
\[ 2x + y = 14 - 3 \]
\[ 2x + y = 11 \]

### Step 6: Substitute the point (-2, p) into the equation
Since the point (-2, p) lies on the line, we substitute x = -2 and y = p into the equation:
\[ 2(-2) + p = 11 \]

### Step 7: Solve for p
Calculating:
\[ -4 + p = 11 \]
Adding 4 to both sides:
\[ p = 11 + 4 \]
\[ p = 15 \]

### Final Answer
The value of p is **15**.

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