If the lines given by `ax^(2)+2hxy+by^(2)=0` are equally inclined to the lines given by `ax^(2)+2hxy+by^(2)+lambda(x^(2)+y^(2))=0`, then

Options

A. `lambda` is any real number
B. `lambda=2`
C. `lambda=1`
D. none of these

Answer

To solve the problem, we need to analyze the two pairs of lines given by the equations:

1. \( ax^2 + 2hxy + by^2 = 0 \) (Equation 1)
2. \( ax^2 + 2hxy + by^2 + \lambda(x^2 + y^2) = 0 \) (Equation 2)

We are tasked with determining the conditions under which these two pairs of lines are equally inclined to one another.

### Step 1: Identify the angle bisector equation for the first pair of lines
The angle bisector equation for the first pair of lines can be derived from the general form of the equation of angle bisectors. The angle bisector equation is given by:

\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} \]

### Step 2: Identify the angle bisector equation for the second pair of lines
For the second pair of lines, we rewrite the equation:

\[ ax^2 + 2hxy + by^2 + \lambda(x^2 + y^2) = 0 \]

This can be rearranged as:

\[ (a + \lambda)x^2 + 2hxy + (b + \lambda)y^2 = 0 \]

The angle bisector equation for this pair of lines is:

\[ \frac{x^2 - y^2}{(a + \lambda) - (b + \lambda)} = \frac{xy}{h} \]

This simplifies to:

\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} \]

### Step 3: Set the angle bisector equations equal
Since the two pairs of lines are equally inclined, their angle bisector equations must be the same. Thus, we equate the two angle bisector equations:

\[ \frac{x^2 - y^2}{a - b} = \frac{xy}{h} \]

This implies that the coefficients of \(x^2\), \(y^2\), and \(xy\) must be consistent across both equations.

### Step 4: Analyze the implications
From our equations, we can see that the angle bisector equations are identical if:

\[ a - b = a + \lambda - (b + \lambda) \]

This simplifies to:

\[ a - b = a - b \]

This holds true for any value of \(\lambda\). Therefore, the lines given by the two equations are equally inclined for any real number value of \(\lambda\).

### Conclusion
Thus, we conclude that the value of \(\lambda\) can be any real number.

### Final Answer
\(\lambda\) is any real number.

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