A point `A` divides the join of `P(-5,1)` and `Q(3,5)` in the ratio `k :1` . Then the integral value of `k` for which the area of ` A B C ,` where `B` is `(1,5)` and `C` is `(7,-2)` , is equal to 2 units in magnitude is___

Explanation

To solve the problem step by step, we will follow the instructions given in the video transcript and apply the section formula and area formula for triangles.

### Step 1: Identify Points and Ratios
We have points:
- \( P(-5, 1) \)
- \( Q(3, 5) \)
- \( B(1, 5) \)
- \( C(7, -2) \)

Point \( A \) divides the line segment \( PQ \) in the ratio \( k:1 \).

### Step 2: Use the Section Formula to Find Coordinates of Point A
Using the section formula, the coordinates of point \( A \) can be calculated as follows:

\[ A\left(\frac{k \cdot x_2 + 1 \cdot x_1}{k + 1}, \frac{k \cdot y_2 + 1 \cdot y_1}{k + 1}\right) \]

Substituting the coordinates of \( P \) and \( Q \):
- \( x_1 = -5, y_1 = 1 \)
- \( x_2 = 3, y_2 = 5 \)

The coordinates of \( A \) become:

\[ A\left(\frac{k \cdot 3 + 1 \cdot (-5)}{k + 1}, \frac{k \cdot 5 + 1 \cdot 1}{k + 1}\right) = A\left(\frac{3k - 5}{k + 1}, \frac{5k + 1}{k + 1}\right) \]

### Step 3: Area of Triangle ABC
The area \( A \) of triangle formed by points \( A, B, C \) can be calculated using the determinant formula:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates:
- \( A\left(\frac{3k - 5}{k + 1}, \frac{5k + 1}{k + 1}\right) \)
- \( B(1, 5) \)
- \( C(7, -2) \)

The area becomes:

\[ \text{Area} = \frac{1}{2} \left| \frac{3k - 5}{k + 1}(5 - (-2)) + 1(-2 - \frac{5k + 1}{k + 1}) + 7(\frac{5k + 1}{k + 1} - 5) \right| \]

This simplifies to:

\[ \text{Area} = \frac{1}{2} \left| \frac{3k - 5}{k + 1} \cdot 7 + 1 \cdot \left(-2 - \frac{5k + 1}{k + 1}\right) + 7 \cdot \left(\frac{5k + 1 - 5(k + 1)}{k + 1}\right) \right| \]

### Step 4: Set the Area Equal to 2
Since the area is given as 2 units, we set up the equation:

\[ \frac{1}{2} \left| \text{Expression} \right| = 2 \]

This leads to:

\[ \left| \text{Expression} \right| = 4 \]

### Step 5: Solve for k
Now we need to solve the expression obtained from the area calculation to find the integral values of \( k \).

After simplifying the expression and solving the resulting equations, we find:

1. \( 14k - 66 = 4k + 4 \) leads to \( k = 7 \)
2. \( 14k - 66 = -4k - 4 \) leads to \( k = \frac{31}{9} \)

Since we are looking for integral values of \( k \), the answer is:

\[ \text{Integral value of } k = 7 \]