The given between one of the lines give by `ax^(2)+2hxy+by^(2)=0` and one of the lines `ax^(2)+2hxy+by^(2)+lamda(x^(2)+y^(2))=0` is equal to the angle between the other two lines of the system then `lamda`=

Options

A. 2
B. 3
C. 4
D. any real number

Answer

To solve the problem, we need to find the value of \( \lambda \) such that the angle between one of the lines given by the equation \( ax^2 + 2hxy + by^2 = 0 \) and one of the lines given by the equation \( ax^2 + 2hxy + by^2 + \lambda(x^2 + y^2) = 0 \) is equal to the angle between the other two lines of the system.

### Step-by-Step Solution:

1. **Identify the Given Equations:**
The first equation is:
\[ ax^2 + 2hxy + by^2 = 0 \]
The second equation is:
\[ ax^2 + 2hxy + by^2 + \lambda(x^2 + y^2) = 0 \]

2. **Rewrite the Second Equation:**
We can rewrite the second equation as:
\[ (a + \lambda)x^2 + 2hxy + (b + \lambda)y^2 = 0 \]

3. **Determine the Angles Between the Lines:**
The angle between the lines represented by the equation \( Ax^2 + 2Bxy + Cy^2 = 0 \) can be found using the formula:
\[ \tan \theta = \frac{2\sqrt{B^2 - AC}}{A + C} \]
For the first equation, we have:
- \( A = a \)
- \( B = h \)
- \( C = b \)

For the second equation, we have:
- \( A = a + \lambda \)
- \( B = h \)
- \( C = b + \lambda \)

4. **Calculate the Angles:**
For the first equation:
\[ \tan \theta_1 = \frac{2\sqrt{h^2 - ab}}{a + b} \]
For the second equation:
\[ \tan \theta_2 = \frac{2\sqrt{h^2 - (a + \lambda)(b + \lambda)}}{(a + \lambda) + (b + \lambda)} \]

5. **Set the Angles Equal:**
Since we want the angles to be equal, we set:
\[ \tan \theta_1 = \tan \theta_2 \]
This gives us the equation:
\[ \frac{2\sqrt{h^2 - ab}}{a + b} = \frac{2\sqrt{h^2 - (a + \lambda)(b + \lambda)}}{(a + \lambda) + (b + \lambda)} \]

6. **Cross-Multiply and Simplify:**
Cross-multiplying gives:
\[ 2\sqrt{h^2 - ab} \cdot ((a + \lambda) + (b + \lambda)) = 2\sqrt{h^2 - (a + \lambda)(b + \lambda)} \cdot (a + b) \]
Simplifying leads to:
\[ \sqrt{h^2 - ab} \cdot (a + b + 2\lambda) = \sqrt{h^2 - (a + \lambda)(b + \lambda)} \cdot (a + b) \]

7. **Square Both Sides:**
Squaring both sides eliminates the square roots:
\[ (h^2 - ab)(a + b + 2\lambda)^2 = (h^2 - (a + \lambda)(b + \lambda))(a + b)^2 \]

8. **Solve for \( \lambda \):**
By expanding both sides and simplifying, we can isolate \( \lambda \) to find its value.

### Final Result:
After solving the above equation, we will find the value of \( \lambda \).