Choose the correct answer from the following :
If the points (p,7), (2,-5) and (6,3) are collinear, then p =

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To determine the value of \( p \) such that the points \( (p, 7) \), \( (2, -5) \), and \( (6, 3) \) are collinear, we can use the condition that the area of the triangle formed by these points must be zero. This can be expressed using the determinant of a matrix formed by the coordinates of the points.

### Step-by-Step Solution:

1. **Set up the determinant**:
The points can be represented as:
- \( (x_1, y_1) = (p, 7) \)
- \( (x_2, y_2) = (2, -5) \)
- \( (x_3, y_3) = (6, 3) \)

The determinant for collinearity is given by:
\[ \begin{vmatrix} p & 7 & 1 \\ 2 & -5 & 1 \\ 6 & 3 & 1 \end{vmatrix} = 0 \]

2. **Calculate the determinant**:
Expanding the determinant, we have:
\[ = p \begin{vmatrix} -5 & 1 \\ 3 & 1 \end{vmatrix} - 7 \begin{vmatrix} 2 & 1 \\ 6 & 1 \end{vmatrix} + 1 \begin{vmatrix} 2 & -5 \\ 6 & 3 \end{vmatrix} \]

Now, calculate each of the 2x2 determinants:
- \( \begin{vmatrix} -5 & 1 \\ 3 & 1 \end{vmatrix} = (-5)(1) - (1)(3) = -5 - 3 = -8 \)
- \( \begin{vmatrix} 2 & 1 \\ 6 & 1 \end{vmatrix} = (2)(1) - (1)(6) = 2 - 6 = -4 \)
- \( \begin{vmatrix} 2 & -5 \\ 6 & 3 \end{vmatrix} = (2)(3) - (-5)(6) = 6 + 30 = 36 \)

Substituting back into the determinant:
\[ = p(-8) - 7(-4) + 36 \]
\[ = -8p + 28 + 36 \]
\[ = -8p + 64 \]

3. **Set the determinant equal to zero**:
To find \( p \), we set the determinant equal to zero:
\[ -8p + 64 = 0 \]

4. **Solve for \( p \)**:
Rearranging gives:
\[ -8p = -64 \]
\[ p = \frac{-64}{-8} = 8 \]

Thus, the value of \( p \) is \( 8 \).