The angle between the lines `ay^2-(1+lambda^2)xy-ax^2=0` is same as the angle between the line:

Options

A. (a) `5x^2+2xy-3y^2=0`
B. (b) `x^2-2xy-3y^2=0`
C. (c) `x^2-y^2=100`
D. (d) xy=0

Answer

To solve the problem, we need to find the angle between the lines represented by the equation \( ay^2 - (1 + \lambda^2)xy - ax^2 = 0 \) and compare it with the angles between the lines represented by the given options.

### Step-by-step Solution:

1. **Identify the coefficients**:
The given equation is \( ay^2 - (1 + \lambda^2)xy - ax^2 = 0 \). We can compare this with the general form of the conic section \( Ax^2 + 2Hxy + By^2 = 0 \).
- Here, \( A = -a \), \( 2H = -(1 + \lambda^2) \) which gives \( H = -\frac{1 + \lambda^2}{2} \), and \( B = a \).

2. **Use the formula for the angle between two lines**:
The formula for the tangent of the angle \( \theta \) between two lines is given by:
\[ \tan \theta = \frac{2\sqrt{H^2 - AB}}{A + B} \]
Substituting the values we found:
- \( A = -a \)
- \( B = a \)
- \( H = -\frac{1 + \lambda^2}{2} \)

3. **Calculate \( H^2 - AB \)**:
\[ H^2 = \left(-\frac{1 + \lambda^2}{2}\right)^2 = \frac{(1 + \lambda^2)^2}{4} \]
\[ AB = (-a)(a) = -a^2 \]
Therefore,
\[ H^2 - AB = \frac{(1 + \lambda^2)^2}{4} + a^2 \]

4. **Calculate \( A + B \)**:
\[ A + B = -a + a = 0 \]
Since \( A + B = 0 \), the angle \( \theta \) is such that \( \tan \theta \) is undefined, which means \( \theta = \frac{\pi}{2} \) or \( 90^\circ \). Thus, the lines are perpendicular.

5. **Check the options**:
We need to find which of the given options also represents lines that are perpendicular.

- **Option A**: \( 5x^2 + 2xy - 3y^2 = 0 \)
- Here, \( A = 5 \), \( B = -3 \) → \( A + B = 5 - 3 = 2 \) (not perpendicular)

- **Option B**: \( x^2 - 2xy - 3y^2 = 0 \)
- Here, \( A = 1 \), \( B = -3 \) → \( A + B = 1 - 3 = -2 \) (not perpendicular)

- **Option C**: \( x^2 - y^2 = 100 \)
- Here, \( A = 1 \), \( B = -1 \) → \( A + B = 1 - 1 = 0 \) (perpendicular)

- **Option D**: \( xy = 0 \)
- This represents the lines \( x = 0 \) and \( y = 0 \). Thus, \( A = 0 \), \( B = 0 \) → \( A + B = 0 \) (perpendicular)

6. **Conclusion**:
The lines represented by options C and D are perpendicular, just like the lines represented by the original equation.

### Final Answer:
The correct options are:
- Option C: \( x^2 - y^2 = 100 \)
- Option D: \( xy = 0 \)