The condition for the lines `lx+my+n=0` and `l_(1)x+m_(1)y+n_(1)=0` to be conjugate with respect to the circle `x^(2)+y^(2)=r^(2)` is

Options

A. `r^(2)(ll_(1)+mm_(1))="nn"_(1)`
B. `r^(2)(ll_(1)-mm_(1))="nn"_(1)`
C. `r^(2)+(ll_(1)+mm_(1))+"nn"_(1)=0`
D. `r^(2)(lm_(1)+ml_(1))="nn"_(1)`

Answer

To find the condition for the lines \( lx + my + n = 0 \) and \( l_1 x + m_1 y + n_1 = 0 \) to be conjugate with respect to the circle \( x^2 + y^2 = r^2 \), we can follow these steps:

### Step 1: Understanding the Concept of Conjugate Lines
Two lines are said to be conjugate with respect to a circle if the pole of one line lies on the other line.

### Step 2: Finding the Pole of the First Line
Let’s denote the first line as \( L_1: lx + my + n = 0 \). The pole of this line with respect to the circle \( x^2 + y^2 = r^2 \) can be found using the formula:
\[ \text{Pole of } L_1 = \left( -\frac{l}{r^2}, -\frac{m}{r^2} \right) \]

### Step 3: Substituting the Pole into the Second Line
Now, we substitute the coordinates of the pole into the second line \( L_2: l_1 x + m_1 y + n_1 = 0 \):
\[ l_1 \left(-\frac{l}{r^2}\right) + m_1 \left(-\frac{m}{r^2}\right) + n_1 = 0 \]
This simplifies to:
\[ -\frac{l l_1 + m m_1}{r^2} + n_1 = 0 \]

### Step 4: Rearranging the Equation
Rearranging the above equation gives:
\[ l l_1 + m m_1 = n_1 r^2 \]

### Step 5: Final Condition for Conjugate Lines
Thus, the condition for the lines to be conjugate with respect to the circle \( x^2 + y^2 = r^2 \) is:
\[ l l_1 + m m_1 - n_1 r^2 = 0 \]

### Step 6: Rewriting the Condition
We can rewrite this condition as:
\[ r^2 (m_1 m + l_1 l) = n_1 \]

### Conclusion
The final condition for the lines \( lx + my + n = 0 \) and \( l_1 x + m_1 y + n_1 = 0 \) to be conjugate with respect to the circle \( x^2 + y^2 = r^2 \) is:
\[ m_1 m r^2 + l_1 l = n_1 \]