If the line Ix + my +n=0 cuts the ellpse `(x^(2))/(a^(2))+(Y^(2))/(b^(2))=1` in point eccentric angles differ by `pi//2`, then

Options

A. `a^(2)l^(2)+b^(2)m^(2)=2n^(2)`
B. `a^(2)l^(2)+b^(2)m^(2)=n^(2)`
C. `a^(2)m^(2)+b^(2)l^(2)=2n^(2)`
D. `a^(2)m^(2)+b^(2)l^(2)=n^(2)`

Explanation

To solve the problem, we need to find the condition under which the line \( lx + my + n = 0 \) intersects the ellipse given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) at points where the eccentric angles differ by \( \frac{\pi}{2} \).

### Step-by-step Solution:

1. **Identify Points on the Ellipse**:
The general point on the ellipse can be represented in terms of the eccentric angle \( \theta \) as:
\[ P = (a \cos \theta, b \sin \theta) \]
The second point, which differs by \( \frac{\pi}{2} \), can be represented as:
\[ Q = (a \cos(\theta + \frac{\pi}{2}), b \sin(\theta + \frac{\pi}{2}) ) \]
Using the trigonometric identities, we find:
\[ Q = (a \cos(\theta + \frac{\pi}{2}), b \sin(\theta + \frac{\pi}{2}) ) = (-a \sin \theta, b \cos \theta) \]

2. **Substituting Points into the Line Equation**:
Since both points \( P \) and \( Q \) lie on the line \( lx + my + n = 0 \), we substitute these points into the line equation.

For point \( P \):
\[ l(a \cos \theta) + m(b \sin \theta) + n = 0 \implies la \cos \theta + mb \sin \theta = -n \tag{1} \]

For point \( Q \):
\[ l(-a \sin \theta) + m(b \cos \theta) + n = 0 \implies -la \sin \theta + mb \cos \theta = -n \tag{2} \]

3. **Squaring and Adding the Two Equations**:
To eliminate \( \theta \), we square both equations and add them:
\[ (la \cos \theta + mb \sin \theta)^2 + (-la \sin \theta + mb \cos \theta)^2 = n^2 + n^2 \]
Expanding both sides:
\[ (la)^2 \cos^2 \theta + 2labm \cos \theta \sin \theta + (mb)^2 \sin^2 \theta + (la)^2 \sin^2 \theta - 2labm \sin \theta \cos \theta + (mb)^2 \cos^2 \theta = 2n^2 \]
Simplifying gives:
\[ (la)^2 (\cos^2 \theta + \sin^2 \theta) + (mb)^2 (\sin^2 \theta + \cos^2 \theta) = 2n^2 \]
Using the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ a^2 l^2 + b^2 m^2 = 2n^2 \]

4. **Final Condition**:
The condition under which the line intersects the ellipse at points with eccentric angles differing by \( \frac{\pi}{2} \) is:
\[ a^2 l^2 + b^2 m^2 = 2n^2 \]