if `h^2=ab` , then the lines represented by `ax^2+2hxy+by^2=0` are

Options

A. Parallel
B. perpendicular
C. coincident
D. None of these

Answer

To solve the problem, we need to analyze the given equation of the pair of straight lines represented by \( ax^2 + 2hxy + by^2 = 0 \) under the condition that \( h^2 = ab \).

### Step-by-Step Solution:

1. **Understand the Equation**: The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of straight lines through the origin. The coefficients \( a \), \( b \), and \( h \) are related to the slopes of these lines.

2. **Condition Given**: We are given the condition \( h^2 = ab \). This condition will help us determine the nature of the lines.

3. **Find the Angle Between the Lines**: The angle \( \theta \) between the two lines can be calculated using the formula:
\[ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| \]

4. **Substituting the Condition**: Since \( h^2 = ab \), we substitute this into the equation:
\[ \tan \theta = \left| \frac{2\sqrt{ab - ab}}{a + b} \right| = \left| \frac{2\sqrt{0}}{a + b} \right| = 0 \]

5. **Interpret the Result**: If \( \tan \theta = 0 \), this implies that the angle \( \theta = 0^\circ \). Therefore, the lines are not just parallel; they actually coincide.

6. **Conclusion**: Since the lines represented by the equation pass through the origin and are coincident, we conclude that the lines are coincident.

### Final Answer:
The lines represented by \( ax^2 + 2hxy + by^2 = 0 \) are **coincident** lines.