If the line \(\mathrm{lx}+\mathrm{my}+\mathrm{n}=0\) cuts an ellipse \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)=1\) in points whose eccentric angles differ by \((\pi / 2)\), then \(\left[\left(a^{2} \ell^{2}+b^{2} \mathrm{~m}^{2}\right) / \mathrm{n}^{2}\right]=\ldots \ldots .\) (a) 1 (b) \((3 / 2)\) (c) 2 (d) \((5 / 2)\)

Understanding the parametric equations of an ellipse is foundational when solving problems related to its geometry. An ellipse is a curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. The standard form of the parametric equations of an ellipse, which has a horizontal major axis, is expressed as:

• \(x = a\cos\theta\)
• \(y = b\sin\theta\)

Here, \(a\) represents the length of the semi-major axis, \(b\) is the length of the semi-minor axis, and \(\theta\) is the parameter, often referred to as the eccentric angle, which varies from 0 to 2\(\pi\). Essentially, these equations delineate the x and y coordinates of any point on the ellipse as the angle \(\theta\) varies. Why parametric? Because unlike the standard (x, y) equation, they allow us to describe the ellipse in terms of a single variable, \(\theta\), which can simplify calculations significantly, especially in problems involving motion and intersection points, as is the case in our exercise.

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. They are incredibly useful in simplifying expressions and solving equations involving trigonometric functions. Some of the most commonly used identities relate the sine and cosine functions and include the Pythagorean identities, such as:

• \(\sin^2\theta + \cos^2\theta = 1\)
• \(1 + \tan^2\theta = \sec^2\theta\)

In the context of our ellipse problem, trigonometric identities are applied to transform the parametric equations for point B, making use of the fact that \(\sin(\theta + \frac{\pi}{2}) = \cos\theta\) and \(\cos(\theta + \frac{\pi}{2}) = -\sin\theta\), to find the coordinates of the point where the eccentric angles differ by \(\pi / 2\). These relationships help us navigate around the unit circle and are crucial for solving many geometry and algebra problems.

The intersection of a line and an ellipse can be determined by substituting the parametric equations of the ellipse into the equation of the line. This process involves equating the x and y coordinates from the ellipse's parametric equations to those in the line's equation, which is a linear equation in the form \(\ell x + my + n = 0\). In our exercise, to find the points where the line intersects the ellipse, we substitute the parametric coordinates into this line equation for both points A and B. It is important to note that the line will cut the ellipse in at most two points. The conditions given in the exercise state that these intersection points have eccentric angles differing by \(\pi/2\), which geometrically means they are separated by a quarter of the ellipse's circumference. By solving the resulting system of equations, we can locate these points and further explore their properties.

To solve ellipse equations, especially those involving intersections with lines, involves a sequence of algebraic manipulations. In the context of the given exercise, after substituting the parametric equations into the line equation and finding two equations representing the intersection points, we need to eliminate the parameter \(\theta\). This is achieved by squaring and adding both equations, harnessing the power of trigonometric identities to simplify expressions. The process results in an equation that relates the constants \(a\), \(b\), \(\ell\), \(m\), and \(n\) without the variable \(\theta\). Such equations are often used to describe the geometric properties of the ellipse or the relationship between the ellipse and intersecting lines. Solving these equations is a matter of skillful manipulation and a strong understanding of both the basic properties of an ellipse and the trigonometric identities.