Statement I . The equation `2x^2-3xy-2y^2+5x-5y+3=0` represents a pair of perpendicular straight lines.
Statement II A pair of lines given by `ax^2+2hxy+by^2+2gx+2fy+c=0` are perpendicular if `a+b=0`

Answer

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

### Step 1: Analyze Statement II
Statement II claims that a pair of lines given by the equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) are perpendicular if \( a + b = 0 \).

1. **Understanding the Condition for Perpendicular Lines**:
- For two lines to be perpendicular, the product of their slopes must equal -1.
- If we denote the slopes of the two lines as \( m_1 \) and \( m_2 \), then the condition for perpendicularity is \( m_1 \cdot m_2 = -1 \).

2. **Finding the Slopes**:
- The slopes can be derived from the coefficients of the quadratic equation.
- If we rewrite the equation in the form of two lines, we can express the slopes in terms of the coefficients \( a_1, b_1 \) and \( a_2, b_2 \).

3. **Condition for Perpendicular Lines**:
- The condition for perpendicular lines can be expressed as:
\[ a_1 a_2 + b_1 b_2 = 0 \]
- If we let \( a = a_1 a_2 \) and \( b = b_1 b_2 \), then for the lines to be perpendicular, we need \( a + b = 0 \).

Thus, Statement II is correct.

### Step 2: Analyze Statement I
Statement I provides the equation \( 2x^2 - 3xy - 2y^2 + 5x - 5y + 3 = 0 \) and claims that it represents a pair of perpendicular straight lines.

1. **Identify Coefficients**:
- From the given equation, we identify:
- Coefficient of \( x^2 \) (denote as \( a \)) = 2
- Coefficient of \( y^2 \) (denote as \( b \)) = -2

2. **Check the Condition for Perpendicular Lines**:
- According to Statement II, for the lines to be perpendicular, we need:
\[ a + b = 0 \]
- Substituting the coefficients:
\[ 2 + (-2) = 0 \]
- This confirms that the condition is satisfied.

Thus, Statement I is also correct.

### Conclusion
Both statements are correct, and Statement II correctly explains Statement I.

### Final Answer:
- Statement I is correct.
- Statement II is correct.
- Statement II is the correct explanation of Statement I.

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