Statement 1: The points `(k+1,k+2),(k,k+1),(k+1,k)` are collinear for any value of k.
Statement 2: If three points are collinear area of the triangle formed by them is zero.

Options

A. Statement-1 is True, Statement-2 is True and Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True and Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To determine whether the points \((k+1, k+2)\), \((k, k+1)\), and \((k+1, k)\) are collinear, we can use the formula for the area of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

If the area is zero, then the points are collinear.

### Step 1: Assign the points
Let:
- \( (x_1, y_1) = (k+1, k+2) \)
- \( (x_2, y_2) = (k, k+1) \)
- \( (x_3, y_3) = (k+1, k) \)

### Step 2: Substitute the coordinates into the area formula
Substituting the coordinates into the area formula gives:

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

### Step 3: Simplify the expression
Now simplify each term inside the absolute value:

1. The first term:
\[ (k+1)(1) = k + 1 \]

2. The second term:
\[ k(k - k - 2) = k(-2) = -2k \]

3. The third term:
\[ (k+1)(1) = k + 1 \]

Combining these, we have:

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

### Step 4: Set the area to zero
For the points to be collinear, we set the area to zero:

\[ \frac{1}{2} \left| 2 - k \right| = 0 \]

This implies:

\[ \left| 2 - k \right| = 0 \]

### Step 5: Solve for \(k\)
Thus, \(2 - k = 0\) or \(k = 2\). This means that the points are only collinear when \(k = 2\). Therefore, the statement that the points are collinear for any value of \(k\) is false.

### Conclusion
- Statement 1 is false: The points are not collinear for all values of \(k\).
- Statement 2 is true: If three points are collinear, the area of the triangle formed by them is zero.

### Final Answer
Statement 1 is false and Statement 2 is true.

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