Problem 61 - A Complete Resource Book in Mathematics for JEE Main

Question

If a chord joining two points whose eccentric angles are \(\alpha, \beta\) cut the major axis of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), at a distance \(d\) from the centre, then \(\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2}=\) (A) \(\frac{d+a}{d-a}\) (B) \(\frac{d-a}{d+a}\) (C) \(\frac{a-d}{a+d}\) (D) none of these

Step-by-Step Solution

Answer

The answer is (B) \(\frac{d-a}{d+a}\).

Step 1: Write Parametric Equations of the Ellipse

The given ellipse is \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\). The parametric equations of an ellipse for eccentric angles \(\alpha\) and \(\beta\) are:\[x_1 = a \cos \alpha, \quad y_1 = b \sin \alpha \x_2 = a \cos \beta, \quad y_2 = b \sin \beta\]

Step 2: Find the Equation of the Chord

The chord joining the points \((x_1, y_1)\) and \((x_2, y_2)\) can be written in the form:\[\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}\]Substituting the parametric forms gives:\[\frac{y - b \sin \alpha}{b \sin \beta - b \sin \alpha} = \frac{x - a \cos \alpha}{a \cos \beta - a \cos \alpha}\]

Step 3: Simplify the Equation of the Chord

Simplify the chord equation to:\((y - b \sin \alpha)(a \cos \beta - a \cos \alpha) = (x - a \cos \alpha)(b \sin \beta - b \sin \alpha)\)After canceling common terms and reorganizing, we have:\(x (b \sin \beta - b \sin \alpha) = y (a \cos \beta - a \cos \alpha) + (ab \sin \alpha \cos \alpha - ab \cos \alpha \sin \alpha)\)

Step 4: Find Point Where Chord Cuts Major Axis

Since the chord cuts the major axis at a distance \(d\) from the center, we know \(y = 0\). This simplifies the equation to:\(x (b \sin \beta - b \sin \alpha) = ab \sin \alpha \cos \alpha - ab \cos \alpha \sin \alpha\)We set this equal to \(ad\) (since distance is \(d\)), solve for \(x\):\(x = \frac{ab (\sin \alpha \cos \beta - \cos \alpha \sin \beta)}{b (\sin \beta - \sin \alpha)}\)

Step 5: Use Trigonometric Identities to Simplify Further

Using the identity \(\sin A \cos B - \cos A \sin B = \sin(A-B)\), simplify:\(x = \frac{ab \sin(\beta - \alpha)}{b(\sin \beta - \sin \alpha)}\)This reduces to:\(x = a \cos \frac{\beta + \alpha}{2}\) means the chord is along the major axis at distance \(d\) or \(-d\).

Step 6: Relate \(x\) to \(d\) and Solve for Eccentric Angles

Since \(x = ad\) or \(x = -ad\), solving the equation gives us conditions for \(\tan \frac{\alpha}{2}\) and \(\tan \frac{\beta}{2}\).Using the identity \(a \cos \frac{\alpha + \beta}{2} = d\), write:\(\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2} = \frac{d-a}{d+a}\), which means the correct choice is:(B) \(\frac{d-a}{d+a}\).

Key Concepts

Eccentric Angles

Eccentric angles are a unique and interesting concept related to ellipses. They help in defining the position of a point on the ellipse in a simpler way. An eccentric angle for a given point on an ellipse is essentially an angular parameter \( \theta \) such that the coordinates \((x, y)\) of the point are expressed in a parametric form based on this angle. This method leverages the cosine and sine functions to relate the distance from the center of the ellipse to the edge at that point:

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

Knowing the eccentric angles, we can easily compute any point on an ellipse using these parametric equations. This is especially useful as it simplifies otherwise complex calculations inherent to the geometry of ellipses.

Chord

A chord is a line segment that joins two points on the curve of an ellipse. Understanding how to find the equation of a chord when the endpoints are given in parametric form is crucial. By using the parametric equations of the ellipse, we can represent points as \((x_1, y_1) = (a \, \cos \alpha, b \, \sin \alpha)\) and \((x_2, y_2) = (a \, \cos \beta, b \, \sin \beta)\). The chord's equation is derived by expressing it in terms of slope and then substituting these parametric expressions:

• Equation format: \(\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}\)

This can be used to explore various geometric properties and intersecting points of the ellipse with other lines and axes.

Parametric Equations of an Ellipse

The parametric equations of an ellipse are fundamental in defining its shape and size. They are derived from the standard equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). The parametric form is especially useful for calculations involving eccentric angles and can be expressed as:

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

With this transformation, any point on the ellipse can be described by an angle \(\theta\), which varies from 0 to \(2\pi\). This method simplifies dealing with ellipses and helps eliminate some of the complexity inherent in Cartesian coordinates.

Major Axis

The major axis of an ellipse is its longest diameter, which stretches across its widest part. It is of significant importance in understanding ellipse geometry. It runs along the x-axis in the standard orientation of the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), and its length is \(2a\). When a chord intersects the major axis, the properties of the ellipse and the chord's equation become relevant in finding specific distances or relationships, such as the distance \(d\) from the center where these intersections occur.

Trigonometric Identities

In the context of ellipses and their geometric properties, trigonometric identities play a crucial role in simplifying expressions. For instance, the identity \(\sin A \cos B - \cos A \sin B = \sin(A-B)\) is instrumental when dealing with expressions involving differences in angles, such as eccentric angles \(\alpha\) and \(\beta\). This identity helps simplify complex expressions and solve problems related to differences or sums of angles. By leveraging these identities, intricate relationships between geometric dimensions and angles on an ellipse can be understood more deeply and resolved efficiently.