The equation `a^2x^2+2h(a+b)x y+b^2y^2=0` and `a x^2+2h x y+b y^2=0` represent

Options

A. two pair of perpendicular straight lines
B. two pairs of parallel straight lines
C. two pairs of straight lines which are equally inclined to each other
D. None of these

Explanation

To determine what the equations \( a^2x^2 + 2h(a+b)xy + b^2y^2 = 0 \) and \( ax^2 + 2hxy + by^2 = 0 \) represent, we can analyze their forms and relationships.

### Step-by-Step Solution:

1. **Identify the Form of the Equations:**
The given equations are quadratic in terms of \( x \) and \( y \). They can be expressed in the general form of a conic section:
\[ Ax^2 + Bxy + Cy^2 = 0 \]
where \( A \), \( B \), and \( C \) are coefficients.

2. **Equation 1:**
The first equation is:
\[ a^2x^2 + 2h(a+b)xy + b^2y^2 = 0 \]
Here, \( A = a^2 \), \( B = 2h(a+b) \), and \( C = b^2 \).

3. **Equation 2:**
The second equation is:
\[ ax^2 + 2hxy + by^2 = 0 \]
Here, \( A = a \), \( B = 2h \), and \( C = b \).

4. **Angle Bisector Equation:**
The angle bisector of two lines represented by the equations can be derived from the general form:
\[ \frac{x^2 - y^2}{A - C} = \frac{xy}{B} \]
For the first equation, we can express it as:
\[ \frac{x^2 - y^2}{a^2 - b^2} = \frac{xy}{2h(a+b)} \]
This can be named as Equation 1.

5. **For the Second Equation:**
Similarly, we can express the second equation as:
\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{2h} \]
This can be named as Equation 2.

6. **Equating the Two Equations:**
Since both equations represent the angle bisector, we equate Equation 1 and Equation 2:
\[ \frac{x^2 - y^2}{a^2 - b^2} = \frac{xy}{2h(a+b)} = \frac{x^2 - y^2}{a - b} = \frac{xy}{2h} \]

7. **Conclusion:**
Since the two equations are equal, it indicates that the lines represented by these equations are equally inclined to each other.

### Final Answer:
The equations \( a^2x^2 + 2h(a+b)xy + b^2y^2 = 0 \) and \( ax^2 + 2hxy + by^2 = 0 \) represent lines that are equally inclined to each other.