Statement I If sum of algebraic distances from points A(1,2),B(2,3),C(6,1) is zero on the line `ax+by+c = 0` then `2a+3b + c = 0` ,
Statement II The centroid of the triangle is (3,2)

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 the statements provided and derive the necessary equations step by step.

### Step 1: Understanding the Problem
We need to check if the sum of algebraic distances from the points A(1,2), B(2,3), and C(6,1) to the line \( ax + by + c = 0 \) being zero implies that \( 2a + 3b + c = 0 \).

### Step 2: Algebraic Distance Formula
The algebraic distance \( d \) from a point \( (x_0, y_0) \) to the line \( ax + by + c = 0 \) is given by:
\[ d = \frac{ax_0 + by_0 + c}{\sqrt{a^2 + b^2}} \]

### Step 3: Setting Up the Equation
For points A(1,2), B(2,3), and C(6,1), the distances can be expressed as:
- Distance from A(1,2):
\[ d_A = \frac{a(1) + b(2) + c}{\sqrt{a^2 + b^2}} = \frac{a + 2b + c}{\sqrt{a^2 + b^2}} \]
- Distance from B(2,3):
\[ d_B = \frac{a(2) + b(3) + c}{\sqrt{a^2 + b^2}} = \frac{2a + 3b + c}{\sqrt{a^2 + b^2}} \]
- Distance from C(6,1):
\[ d_C = \frac{a(6) + b(1) + c}{\sqrt{a^2 + b^2}} = \frac{6a + b + c}{\sqrt{a^2 + b^2}} \]

### Step 4: Summing the Distances
The sum of the algebraic distances is:
\[ d_A + d_B + d_C = \frac{a + 2b + c + 2a + 3b + c + 6a + b + c}{\sqrt{a^2 + b^2}} = \frac{(1 + 2 + 6)a + (2 + 3 + 1)b + 3c}{\sqrt{a^2 + b^2}} \]
This simplifies to:
\[ \frac{9a + 6b + 3c}{\sqrt{a^2 + b^2}} = 0 \]

### Step 5: Setting the Numerator to Zero
For the sum to be zero, the numerator must be zero:
\[ 9a + 6b + 3c = 0 \]
Dividing the entire equation by 3 gives:
\[ 3a + 2b + c = 0 \]

### Step 6: Analyzing Statement I
The statement claims that if the sum of distances is zero, then \( 2a + 3b + c = 0 \). However, we derived \( 3a + 2b + c = 0 \). Therefore, Statement I is **false**.

### Step 7: Finding the Centroid of Triangle ABC
The centroid \( G \) of triangle ABC with vertices A(1,2), B(2,3), and C(6,1) is given by:
\[ G_x = \frac{x_1 + x_2 + x_3}{3} = \frac{1 + 2 + 6}{3} = \frac{9}{3} = 3 \]
\[ G_y = \frac{y_1 + y_2 + y_3}{3} = \frac{2 + 3 + 1}{3} = \frac{6}{3} = 2 \]
Thus, the centroid is \( (3, 2) \), confirming Statement II is **true**.

### Conclusion
- Statement I: False
- Statement II: True

### Final Answer
The correct option is that Statement I is false and Statement II is true.