Statement 1 : The chord of contact of the circle `x^2+y^2=1` w.r.t. the points (2, 3), (3, 5), and (1, 1) are concurrent. Statement 2 : Points (1, 1), (2, 3), and (3, 5) are collinear.

Options

A. Statement I is true, Statement II is true, Statement II is a correct explanation for Statement I
B. Statement I is true, Statement II is true, Statement II is not a correct explanation for Statement I
C. Statement I is true, Statement II is false
D. Statement I is false, Statement II is true

Answer

To solve the problem, we need to analyze both statements provided and determine their validity.

### Step 1: Analyze Statement 2
**Statement 2**: Points (1, 1), (2, 3), and (3, 5) are collinear.

To check if these points are collinear, we can calculate the slopes between the points.

1. **Calculate the slope between points (1, 1) and (2, 3)**:
\[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{2 - 1} = \frac{2}{1} = 2 \]

2. **Calculate the slope between points (2, 3) and (3, 5)**:
\[ \text{slope}_{BC} = \frac{5 - 3}{3 - 2} = \frac{2}{1} = 2 \]

Since both slopes are equal, the points (1, 1), (2, 3), and (3, 5) are collinear.

**Conclusion for Statement 2**: True.

### Step 2: Analyze Statement 1
**Statement 1**: The chord of contact of the circle \(x^2 + y^2 = 1\) with respect to the points (2, 3), (3, 5), and (1, 1) are concurrent.

To determine if the chord of contact is concurrent, we can use the concept of the determinant.

1. **Set up the determinant**:
We will use the points (2, 3), (3, 5), and (1, 1) to form the determinant:
\[ \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} = \begin{vmatrix} 2 & 3 & 1 \\ 3 & 5 & 1 \\ 1 & 1 & 1 \end{vmatrix} \]

2. **Calculate the determinant**:
\[ = 2(5 - 1) - 3(3 - 1) + 1(3 - 5) \]
\[ = 2 \cdot 4 - 3 \cdot 2 + 1 \cdot (-2) \]
\[ = 8 - 6 - 2 = 0 \]

Since the determinant equals 0, the points are concurrent.

**Conclusion for Statement 1**: True.

### Final Conclusion
Both statements are true, and Statement 2 correctly explains Statement 1.

### Answer
**Option 1**: Statement 1 is true, Statement 2 is true, and Statement 2 is the correct explanation for Statement 1.

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