Problem 40 - A Complete Resource Book in Mathematics for JEE Main

Question

If the eccentric angle of a point lying in the first quadrant on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) be \(\alpha\) and the line joining the centre to the point makes an angle \(\beta\) with \(x\)-axis then \(\alpha-\beta\) will be maximum when \(\alpha=\) (A) 0 (B) \(\cot ^{-1} \sqrt{\frac{a}{b}}\) (C) \(\tan ^{-1} \sqrt{\frac{a}{b}}\) (D) \(\pi / 4\)

Step-by-Step Solution

Answer

The maximum occurs when \(\alpha = \tan^{-1} \sqrt{\frac{a}{b}}\).

Step 1: Understand the Eccentric Angle

For the ellipse defined by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the parametric equations are \(x = a\cos\alpha\) and \(y = b\sin\alpha\), where \(\alpha\) is the eccentric angle. This angle helps express points on the ellipse in a cartesian form.

Step 2: Express the Angle \(\beta\)

The angle \(\beta\) made with the x-axis by the line joining the center of the ellipse (0,0) to the point \((x, y)\) is given by \(\beta = \tan^{-1}\left(\frac{y}{x}\right)\). Using the parametric equations, \(\beta = \tan^{-1}\left(\frac{b\sin\alpha}{a\cos\alpha}\right)\).

Key Concepts

Ellipse

An ellipse is one of the most fascinating shapes in geometry, belonging to the family of conic sections. It looks like an elongated circle and is often defined by its parametric equations or standard form: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) are the semi-major and semi-minor axes, respectively.
The ellipse has a specific center, which is usually considered the origin in a coordinate system. Every ellipse exhibits two focal points located along its major axis. The sum of the distances from these focal points to any point on the ellipse is constant. This unique characteristic distinguishes an ellipse from other conic sections like circles or hyperbolas.
Ellipses are used in various applications such as astronomy, physics, and engineering. They approximate orbits of planets and satellites and are used in architectural designs for acoustics.

Parametric Equations

Parametric equations are a powerful tool in mathematics that allow us to represent curves and shapes like ellipses in terms of a single parameter, such as \(\alpha\) in the context of the ellipse. For an ellipse, the parametric equations are given by \(x = a\cos\alpha\) and \(y = b\sin\alpha\).
The parameter \(\alpha\) is known as the eccentric angle, crucial for locating any point on the ellipse. By substituting different values for \(\alpha\), one can determine the coordinates \((x, y)\) of various points on the ellipse.
These equations simplify calculations involved in analyzing curves, making it easier to compute intersections, tangents, and other geometric properties. Parametric equations are widely utilized in computer graphics, simulations, and designing animations.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are fundamental in the study of parametric equations, particularly concerning ellipses. In the equations \(x = a\cos\alpha\) and \(y = b\sin\alpha\), sine and cosine describe rotation and scaling transformations.
These functions help convert angular measures into linear coordinates, enabling the translation of points as the angle \(\alpha\) varies. Understanding how the functions relate angles to the sides of a right-angled triangle provides a foundation for grasping more complex geometric shapes.
By employing trigonometric identities, such as the Pythagorean identity \(\cos^2\alpha + \sin^2\alpha = 1\), we can verify and validate relationships within geometric figures like ellipses. They are essential for solving trigonometric equations and leveraging them in real-world applications like signal processing and mechanical simulations.

Geometry of Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. The geometry of conic sections includes common shapes like circles, ellipses, parabolas, and hyperbolas. Each has unique properties and equations governing their forms.
Ellipses, specifically, are formed when the intersecting plane is angled such that it cuts through both halves of the cone but not perpendicularly, resulting in a closed symmetrical shape. Their standard equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) describes the locus of points satisfying specific distance criteria from the foci.
Understanding the geometry of conic sections helps in various domains, including planetary and orbital mechanics, whereby ellipses are used to describe the paths of celestial bodies. Conic sections also appear in optics, design, and structural engineering, influencing how light, sound, and stress patterns behave.