The lines represented by `x^(2)+2lambda xy+2y^(2)=0` and the lines represented by `1+lambda_x^(2)-8xy+y^(2)=0` are equally inclined, then

Options

A. `lambda` is any real number
B. `lambda gt 2`
C. `lambda = pm2`
D. `lambda lt -2`

Explanation

To solve the problem, we need to analyze the given equations of the lines and determine the conditions under which they are equally inclined.

### Step-by-Step Solution:

1. **Identify the equations of the lines:**
The first equation is:
\[ x^2 + 2\lambda xy + 2y^2 = 0 \]
The second equation is:
\[ 1 + \lambda x^2 - 8xy + y^2 = 0 \]

2. **Rewrite the first equation:**
The first equation can be rewritten in the form of a quadratic in \(x\) and \(y\):
\[ x^2 + 2\lambda xy + 2y^2 = 0 \]
This represents two lines which can be expressed as:
\[ m_1, m_2 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \(a = 1\), \(b = 2\lambda\), and \(c = 2\).

3. **Calculate the slopes of the lines:**
The slopes of the lines represented by the first equation are given by:
\[ m_1, m_2 = \frac{-2\lambda \pm \sqrt{(2\lambda)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]
Simplifying this gives:
\[ m_1, m_2 = \frac{-2\lambda \pm \sqrt{4\lambda^2 - 8}}{2} \]
\[ = -\lambda \pm \sqrt{\lambda^2 - 2} \]

4. **Rewrite the second equation:**
The second equation can be rearranged as:
\[ \lambda x^2 - 8xy + (y^2 + 1) = 0 \]
This also represents two lines.

5. **Calculate the slopes of the lines from the second equation:**
The slopes of the lines represented by the second equation are given by:
\[ m_3, m_4 = \frac{8 \pm \sqrt{(-8)^2 - 4\lambda(1)}}{2\lambda} \]
\[ = \frac{8 \pm \sqrt{64 - 4\lambda}}{2\lambda} \]

6. **Condition for equally inclined lines:**
For the lines to be equally inclined, the product of the slopes must equal -1:
\[ (m_1 - m_2)(m_3 - m_4) = -1 \]

7. **Set up the equations:**
From the slopes calculated, we can set up the equations:
\[ (-\lambda + \sqrt{\lambda^2 - 2}) - (-\lambda - \sqrt{\lambda^2 - 2}) = 2\sqrt{\lambda^2 - 2} \]
and
\[ (8 + \sqrt{64 - 4\lambda}) - (8 - \sqrt{64 - 4\lambda}) = 2\sqrt{64 - 4\lambda} \]

8. **Equate and solve for \(\lambda\):**
Setting up the equation:
\[ 2\sqrt{\lambda^2 - 2} \cdot 2\sqrt{64 - 4\lambda} = -1 \]
Simplifying this will lead to a quadratic equation in \(\lambda\).

9. **Final result:**
Solving the quadratic equation will yield the values of \(\lambda\) that satisfy the condition of equal inclination.