The equation `ax^(2)=2hxy+by^(2)=0` represented a pair of coincident lines through the origin if

Options

A. `h^(2)=ab`
B. `2h=ab`
C. `a=bh`
D. `b=ah`

Answer

To determine the condition under which the equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of coincident lines through the origin, we can follow these steps:

### 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. For these lines to be coincident, they must be identical.

2. **Condition for Coincident Lines**:
For the lines represented by the quadratic equation to be coincident, the angle \( \theta \) between the lines must be zero. This means that the tangent of the angle \( \theta \) must also be zero.

3. **Using the Formula for Tangent of Angle**:
The formula for the tangent of the angle \( \theta \) between the two lines represented by the equation is given by:
\[ \tan \theta = \pm \sqrt{\frac{2h^2 - ab}{a + b}} \]
For the lines to be coincident, we set \( \tan \theta = 0 \).

4. **Setting Up the Equation**:
Setting \( \tan \theta = 0 \) implies:
\[ \sqrt{2h^2 - ab} = 0 \]
This leads us to:
\[ 2h^2 - ab = 0 \]

5. **Solving for the Condition**:
Rearranging the equation gives us:
\[ 2h^2 = ab \]
Thus, the condition for the equation to represent a pair of coincident lines is:
\[ h^2 = \frac{ab}{2} \]

### Conclusion:
The equation \( ax^2 + 2hxy + by^2 = 0 \) represents a pair of coincident lines through the origin if \( h^2 = \frac{ab}{2} \).