The line `lx+my+n=0` intersects the curve `ax^2 + 2hxy + by^2 = 1` at the point P and Q. The circle on PQ as diameter passes through the origin. Then `n^2(a+ b)` equals (A) `l^2+m^2` (B) `2lm` (C) `l^2-m^2` (D) `4lm`

Options

A. `n^(2)(a+b)`
B. `n^(2)(a+b)^(2)`
C. `n^(2)(a^(2)-b^(2))`
D. `n^(2)(a^(2)+b^(2))`

Answer

To solve the problem, we need to find the value of \( n^2(a + b) \) given the line \( lx + my + n = 0 \) intersects the curve \( ax^2 + 2hxy + by^2 = 1 \) at points P and Q, and that the circle with diameter PQ passes through the origin.

### Step-by-Step Solution:

1. **Rewrite the line equation**:
The line can be rewritten as:
\[ lx + my = -n \]

2. **Homogenize the curve with the line**:
We will homogenize the given curve with the line. This means we will express the curve in terms of the line's coefficients. The equation becomes:
\[ ax^2 + 2hxy + by^2 = \frac{(lx + my)^2}{n^2} \]
This leads to:
\[ ax^2 + 2hxy + by^2 - \frac{(lx + my)^2}{n^2} = 0 \]

3. **Simplify the equation**:
Expanding the right side:
\[ ax^2 + 2hxy + by^2 - \frac{l^2x^2 + 2lmyx + m^2y^2}{n^2} = 0 \]
Rearranging gives:
\[ \left(a - \frac{l^2}{n^2}\right)x^2 + \left(2h - \frac{2lm}{n^2}\right)xy + \left(b - \frac{m^2}{n^2}\right)y^2 = 0 \]

4. **Identify the roots**:
This quadratic equation in \( x \) and \( y \) has roots corresponding to the slopes of the lines through points P and Q. Let the slopes be \( m_1 \) and \( m_2 \).

5. **Use the property of slopes**:
Since the circle with diameter PQ passes through the origin, the product of the slopes \( m_1 \) and \( m_2 \) must equal -1:
\[ m_1 m_2 = -1 \]
From the quadratic, we have:
\[ m_1 m_2 = \frac{a - \frac{l^2}{n^2}}{b - \frac{m^2}{n^2}} = -1 \]

6. **Set up the equation**:
This gives us:
\[ a - \frac{l^2}{n^2} = -\left(b - \frac{m^2}{n^2}\right) \]
Rearranging leads to:
\[ a + b = \frac{l^2 + m^2}{n^2} \]

7. **Multiply by \( n^2 \)**:
Multiplying both sides by \( n^2 \):
\[ n^2(a + b) = l^2 + m^2 \]

8. **Final result**:
Therefore, we conclude:
\[ n^2(a + b) = l^2 + m^2 \]

### Conclusion:
The value of \( n^2(a + b) \) is \( l^2 + m^2 \).