If the two pairs of line `x^2 -2mxy -y^2=0 and x^2 - 2nxy -y^2 = 0` are such that one of them represent the bisector of the angles between the other, then: (A) mn + 1 = 0 (B) mn - 1 = 0 (C) 1/m + 1/n = 0 (D) 1/m -1/n = 0

Options

A. `mn+1=0`
B. `mn-1=0`
C. `(1)/(m)+(1)/(n)=0`
D. `(1)/(m)-(1)/(n)=0`

Answer

To solve the problem, we need to analyze the given equations of the pairs of lines and determine the relationship between \( m \) and \( n \) under the condition that one pair represents the angle bisector of the other.

### Step-by-Step Solution:

1. **Write the Given Equations:**
The two pairs of lines are given by:
\[ x^2 - 2mxy - y^2 = 0 \quad \text{(1)} \]
\[ x^2 - 2nxy - y^2 = 0 \quad \text{(2)} \]

2. **Identify the Angle Bisector Condition:**
We need to find the angle bisector of the lines represented by equation (2). The angle bisector of two lines given by the general form \( Ax^2 + Bxy + Cy^2 = 0 \) can be derived using the formula:
\[ \frac{x^2 - y^2}{A - C} = \frac{2xy}{B} \]
For our case, we have:
- \( A = 1 \)
- \( B = -2n \)
- \( C = -1 \)

3. **Apply the Angle Bisector Formula:**
The angle bisector equation for the second pair of lines becomes:
\[ \frac{x^2 - y^2}{1 - (-1)} = \frac{2xy}{-2n} \]
Simplifying this gives:
\[ x^2 - y^2 = -\frac{2xy}{-2n} \implies x^2 - y^2 = \frac{xy}{n} \]
Rearranging, we get:
\[ x^2 + \frac{2xy}{n} - y^2 = 0 \quad \text{(3)} \]

4. **Equate the Two Equations:**
Since equation (1) is the angle bisector of equation (2), we can equate (1) and (3):
\[ x^2 - 2mxy - y^2 = x^2 + \frac{2xy}{n} - y^2 \]
Cancelling \( x^2 \) and \( -y^2 \) from both sides gives:
\[ -2mxy = \frac{2xy}{n} \]

5. **Simplify the Equation:**
Dividing both sides by \( xy \) (assuming \( xy \neq 0 \)):
\[ -2m = \frac{2}{n} \]
Rearranging this gives:
\[ mn + 1 = 0 \]

6. **Conclusion:**
Thus, we have derived the condition:
\[ mn + 1 = 0 \]
Therefore, the correct option is:
\[ \text{(A) } mn + 1 = 0 \]