For the second degree equation `ax^(2)+2hxy+by^(2)+2gx+2fy+c=0`
which of the following is not true ?

Options

A. represent a pair of straight line if `Delta=|(a,h,g),(h,b,f),(g,f,c)|=0`
B. represent a pair of perpendicular lines if `Delta=0` and `a+b=0`
C. represent a rectangular hyperbola if `Deltane0,a+b=0 and h^(2)gtab`
D. none

Answer

To solve the problem regarding the second-degree equation \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \) and identify which statement is not true, we will analyze each option based on the properties of conic sections.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The given equation is a general second-degree equation in two variables \( x \) and \( y \). It can represent different types of conic sections depending on the values of the coefficients \( a, b, h, g, f, \) and \( c \).

2. **Identifying Conditions for Conic Sections**:
- The discriminant \( \Delta \) for the conic section is given by:
\[ \Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} \]
- The nature of the conic section can be determined based on the value of \( \Delta \):
- If \( \Delta = 0 \), the conic represents a pair of straight lines.
- If \( \Delta > 0 \), it represents an ellipse or a circle.
- If \( \Delta < 0 \), it represents a hyperbola.

3. **Analyzing Each Option**:
- **Option 1**: The equation represents a pair of straight lines if \( \Delta = 0 \).
- This statement is **true**.

- **Option 2**: The equation represents a pair of perpendicular lines.
- For two lines to be perpendicular, the condition \( a + b = 0 \) must hold true along with \( \Delta = 0 \). This statement is also **true**.

- **Option 3**: The equation represents a rectangular hyperbola.
- A rectangular hyperbola is characterized by the conditions \( \Delta \neq 0 \), \( a + b = 0 \), and \( h^2 > ab \). This statement is **true**.

- **Option 4**: None of the above statements is false.
- Since all previous statements are true, this statement is also **true**.

4. **Conclusion**:
Since all the options provided are true, we conclude that there is no false statement among the options given. Thus, the answer to the question "which of the following is not true?" is that **none of the options are false**.