Statement I. The four straight lines given by
`6x^2+5xy-6y^2=0 and 6x^2+5xy-6y^2-x+5y-1=0` are the sides of a square .
Statement II . The lines represented by general equation of second degree `ax^2+2hxy+by^2+2gx+2fy+c=0 ` are perpendicular if a+b=0 .

Options

A. A. Statement I is true, Statement II is true , Statement II is a correct explanation for Statement I
B. B. Statement I is true , Statement II is true , Statement II is not a correct explanation for statement I
C. C. Statement I is true , Statement II is false
D. 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 II
The second statement claims that the lines represented by the general equation of the second degree \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) are perpendicular if \( a + b = 0 \).

1. **Understanding the Condition for Perpendicular Lines**:
- The angle \( \theta \) between the lines can be determined using the formula:
\[ \tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b} \]
- For the lines to be perpendicular, \( \tan \theta \) must be undefined, which occurs when \( a + b = 0 \).

2. **Conclusion for Statement II**:
- Since \( a + b = 0 \) results in the lines being perpendicular, Statement II is **true**.

### Step 2: Analyze Statement I
The first statement claims that the four straight lines given by \( 6x^2 + 5xy - 6y^2 = 0 \) and \( 6x^2 + 5xy - 6y^2 - x + 5y - 1 = 0 \) are the sides of a square.

1. **Finding the Lines from the First Equation**:
- The equation \( 6x^2 + 5xy - 6y^2 = 0 \) can be factored to find the slopes of the lines.
- This is a quadratic in \( x \) and can be expressed as:
\[ 6x^2 + 5xy - 6y^2 = 0 \implies (6x - 6y)(x + y) = 0 \]
- The lines are \( y = x \) and \( y = -\frac{6}{5}x \).

2. **Finding the Lines from the Second Equation**:
- The second equation \( 6x^2 + 5xy - 6y^2 - x + 5y - 1 = 0 \) can also be analyzed similarly.
- Rearranging gives us:
\[ 6x^2 + 5xy - 6y^2 = x - 5y + 1 \]
- This is more complex, but we can check if the slopes yield perpendicular lines.

3. **Checking Perpendicularity**:
- For the lines to form a square, they must be perpendicular.
- We check the coefficients of \( x^2 \) and \( y^2 \) from both equations:
- From the first equation, \( a_1 = 6 \) and \( b_1 = -6 \) gives \( a_1 + b_1 = 0 \).
- From the second equation, we can similarly find \( a_2 + b_2 = 0 \).

4. **Conclusion for Statement I**:
- Since both sets of lines are perpendicular, they can indeed form the sides of a square. Thus, Statement I is **true**.

### Final Conclusion
Both statements are true.