Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Option (B): \(\frac{d-a}{d+a}\)

Step 1: Identify Coordinates of Points on the Ellipse

For an ellipse described by the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), any point on the ellipse with eccentric angle \(\theta\) can be expressed as \((a \cos \theta, b \sin \theta)\). Consequently, the points corresponding to eccentric angles \(\alpha\) and \(\beta\) are \((a \cos \alpha, b \sin \alpha)\) and \((a \cos \beta, b \sin \beta)\) respectively.

Step 2: Derive the Equation of the Chord

The equation of the chord joining points \((a \cos \alpha, b \sin \alpha)\) and \((a \cos \beta, b \sin \beta)\) is derived from the line segment formula: \(\frac{x}{a \cos \alpha} + \frac{y}{b \sin \alpha} = 1\). However, to find its relevance for the distance \(d\), it is often more useful to equate the chord using the parametric form: \(y = m(x - c)\), where \(c = \frac{a(\cos \alpha + \cos \beta)}{2}\) is on the x-axis.

Step 3: Calculate Distance from the Centre to the Chord

Since the chord cuts the major axis at a distance \(d\), this implies the perpendicular distance from the centre (origin) to the chord is \(d\). The perpendicular distance from the origin to a line defined in the general form \(Ax + By + C = 0\) is \(\frac{|C|}{\sqrt{A^2 + B^2}}\). For our derived line equation, solve to find this distance and equate it to \(d\).

Step 4: Apply Trigonometric Identity for Solution Simplification

Relate the previous expression to trigonometric identities for tangent. Using double angle formulas, the expression \(\tan(\alpha/2) \cdot \tan(\beta/2)\) can be simplified using identities such as \(\cos \alpha = \frac{1 - \tan^2(\alpha/2)}{1 + \tan^2(\alpha/2)}\) and similar for \(\cos \beta\).

Step 5: Conclude with Expression for Tangent Product

Ultimately, the trigonometric manipulations should lead to an expression where \(\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2} = \frac{d-a}{d+a}\). This matches with option (B) from the choices.

Key Concepts

Eccentric Angles

In the context of ellipses, eccentric angles are a way of parameterizing the points on the ellipse. If we have an ellipse represented by the equation \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\), each point on this ellipse can be described using an eccentric angle \(\theta\). An eccentric angle forms between the major axis of the ellipse and a line from the center of the ellipse to a point \((x, y)\) on the ellipse.

• These angles help in expressing positions on the ellipse in a simplified form.
• The formula used to determine the coordinates of a point on the ellipse is \((a \cos \theta, b \sin \theta)\).

In practical terms, by knowing the eccentric angle, we can easily determine the position of a specific point on the ellipse. It simplifies the process of working with ellipses, especially when dealing with complex geometric problems like finding chords and tangents related to the ellipse.

Chord of an Ellipse

A chord in the context of an ellipse is a straight line segment that joins two points on the ellipse. The equation of a chord can be derived using the coordinates of two distinct points on the ellipse. For points associated with eccentric angles \(\alpha\) and \(\beta\), the coordinates will be \((a \cos \alpha, b \sin \alpha)\) and \((a \cos \beta, b \sin \beta)\), respectively.

• To find the equation of the chord, the line segment formula is often employed.
• Alternatively, a parametric form such as \(y = m(x - c)\) can be more straightforward for some analyses.
• The midpoint formula for the chord can also be useful. In the case of the minor axis being cut, this point lies on the major axis.

Understanding chords is key in many applications involving ellipses, particularly when the chord's intersection properties with the axes are in question. This leads naturally to discussions about distances from the center and how the chord impacts the geometry.

Tangent Product Identity

The tangent product identity involves determining special trigonometric relationships that can simplify expressions with eccentric angles. Specifically, for an ellipse, you might encounter tasks that require finding the product of tangents of half-eccentric angles, \(\tan(\alpha/2)\) and \(\tan(\beta/2)\).

• Using these identities can transform complex trigonometric equations into simpler forms.
• One useful identity involves double angle formulas, such as \(\cos \alpha = \frac{1 - \tan^2(\alpha/2)}{1 + \tan^2(\alpha/2)}\).
• These simplifications can relate the distances involved, leading to clear solutions like \(\tan \frac{\alpha}{2} \cdot \tan \frac{\beta}{2} = \frac{d-a}{d+a}\).

The tangent product identity is useful for deriving results in problems dealing with ellipses, such as when you need to compute specific values related to chord distances on the axes. Understanding these identities allows for efficient and accurate solutions in geometric contexts involving ellipses.