Statement-1: All chords of the curve `3x^(2)-y^(2)-2x+4y=0` which subtend a right angle at the origin pass through a fixed point.
Statement-2: The equation `ax+by+c=0` represents a family of straight lines passing through a fixed point iff there is a linear relation between a, b and c.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, 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.

Explanation

To solve the problem, we need to analyze the statements provided and derive the necessary conclusions step-by-step.

### Step 1: Analyze the Curve
The given curve is:
\[ 3x^2 - y^2 - 2x + 4y = 0 \]

### Step 2: Rewrite the Curve
We can rearrange the equation to understand its structure better:
\[ 3x^2 - 2x - y^2 + 4y = 0 \]

### Step 3: Identify the Chords
A chord of the curve that subtends a right angle at the origin can be represented by the equation:
\[ ax + by = 1 \]
where \( a \) and \( b \) are constants.

### Step 4: Substitute the Chord Equation into the Curve
Substituting \( y \) from the chord equation into the curve equation:
1. From \( ax + by = 1 \), we can express \( y \) as:
\[ y = \frac{1 - ax}{b} \]

2. Substitute this expression for \( y \) into the curve equation:
\[ 3x^2 - \left(\frac{1 - ax}{b}\right)^2 - 2x + 4\left(\frac{1 - ax}{b}\right) = 0 \]

### Step 5: Simplify the Equation
Expanding and simplifying the above equation will yield a quadratic in \( x \). The condition for the lines represented by this quadratic to be at right angles is given by the determinant condition.

### Step 6: Condition for Right Angles
For the lines to be at right angles, the following condition must hold:
\[ 3 - 2a + 4b - 1 = 0 \]
This simplifies to:
\[ a - 2b - 1 = 0 \]

### Step 7: Fixed Point
The equation \( a - 2b - 1 = 0 \) represents a linear relationship between \( a \) and \( b \). According to Statement 2, if there is a linear relation between \( a \), \( b \), and \( c \) (where \( c \) can be considered as a constant in the equation of the line), then all such lines pass through a fixed point.

### Conclusion
Thus, we conclude that:
- Statement 1 is true: All chords of the curve that subtend a right angle at the origin pass through a fixed point.
- Statement 2 is also true: The equation \( ax + by + c = 0 \) represents a family of straight lines passing through a fixed point if there is a linear relation between \( a \), \( b \), and \( c \).

### Final Answer
Both statements are true, and Statement 2 is a correct explanation for Statement 1.

---