Problem 39 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the eccentric angles of the ends of a focal chord of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b)\) are \(\theta_{1}\) and \(\theta_{2}\), then value of \(\tan \theta_{1} \tan \theta_{2}\) equals (A) \(\frac{e-1}{e+1}\) (B) \(\frac{e-1}{e^{2}+1}\) (C) \(\frac{e+1}{e-1}\) (D) \(\frac{e^{2}+1}{e-1}\)

Step-by-Step Solution

Answer

The correct option is (A) \( \frac{e-1}{e+1} \).

Step 1: Understand the Problem Statement

We are given an ellipse described by the equation \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b \). A focal chord is a line segment passing through the focus of the ellipse and having endpoints on the ellipse. We are to find the expression for \( \tan \theta_1 \tan \theta_2 \) where \( \theta_1 \) and \( \theta_2 \) are the eccentric angles of the endpoints of the focal chord.

Step 2: Use the Equation of the Ellipse

The parametric equations of an ellipse are \( x = a \cos \theta \) and \( y = b \sin \theta \). This will be useful for calculating the coordinates of the endpoints of a chord for given \( \theta_1 \) and \( \theta_2 \).

Step 3: Recall Focal Chord Property

In an ellipse, the product of \( \tan \theta_1 \tan \theta_2 \) for a focal chord is a property tied to the eccentricity \( e \), where \( e = \sqrt{1 - \frac{b^2}{a^2}} \). This property simplifies our calculation.

Step 4: Identify Focal Chord Formula

For a focal chord of an ellipse, the identity \( \tan \theta_1 \tan \theta_2 = \frac{b^2}{a^2} \) applies directly. Simplifying with \( e \), where \( e^2 = 1 - \frac{b^2}{a^2} \).

Step 5: Apply Eccentricity Simplification

Re-write \( \frac{b^2}{a^2} \) in terms of eccentricity \( e \):\[ \frac{b^2}{a^2} = 1 - e^2 \]Then, simplify to identify the correct choice among the options given.

Step 6: Choose the Correct Option

Substitute \( 1 - e^2 = \frac{b^2}{a^2} \) into \( \tan \theta_1 \tan \theta_2 = 1 - e^2 \):This reduces to the form given in the option (A) \( \frac{e-1}{e+1} \).

Key Concepts

Eccentric Angle

In the context of an ellipse, the eccentric angle \( \theta \) is associated with a point on the ellipse. It provides a parametric representation of that point on theellipse. Consider an ellipse given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where a > b. Here, \( a \) is the semi-major axis and \( b \) is the semi-minor axis.For any point \((x, y)\) on the ellipse, we can express:- \( x = a \cos \theta \)- \( y = b \sin \theta \)This shows how the position of a point is parameterized by the angle \( \theta \).These expressions make it easy to establish relationships between differentpoints on the ellipse. When we use eccentric angles for points at the ends of a focal chord—a line segment going through the focus with both ends on the ellipse—it becomes simpler to analyze properties such as the product of tangents of these angles.

Focal Chord

A focal chord of an ellipse is a line segment that passes through one of thefoci of the ellipse, extending outwards to intersect the ellipse at bothends. This particular property is key in problems involving eccentric angles.Imagine an ellipse with a focus at point \((ae, 0)\), where \(e\) represents the eccentricity.The focal chord not only highlights symmetry in the ellipse's geometry, butalso ties to the ellipse's focal properties.

Key Properties of Focal Chords

- **Symmetry**: A focal chord highlights symmetry within the structure of an ellipse by intersecting both the arc and foci.- **Tangent Relationships**: A focal chord can help identify trigonometric relationships, such as \( \tan \theta_1 \tan \theta_2 \) for eccentric angles \( \theta_1 \) and \( \theta_2 \).- **Focal Distance**: The length of a focal chord can be determined using properties of eccentricity \(e\), further connecting the chord to other ellipse elements.Understanding these highlight how focal chords are crucial in solving many geometric problems involving ellipses.

Eccentricity

Eccentricity \(e\) is a fundamental property of ellipses that describes how much an ellipse deviates from being a circle. It is defined for an ellipse with semi-major axis \(a\) and semi-minor axis \(b\) as:\[ e = \sqrt{1 - \frac{b^2}{a^2}} \]The value of \(e\) provides a measure of the "flatness" of the ellipse:- If \(e = 0\), the ellipse is a circle.- If \(0 < e < 1\), the ellipse retains its oval shape.

Importance in Ellipse Problems

- **Geometry**: Eccentricity helps describe the shape and geometry of the ellipse, clearly indicating its elongation.- **Focal Length**: Eccentricity connects directly to the distance between the foci of the ellipse, \(ae\), and thus helps in calculating the focal chord length.- **Trigonomic Properties**: Many identities and relationships involving the ellipse’s geometry, like those with eccentric angles and focal chords, depend on knowing \(e\).Because of these features, understanding eccentricity is crucial for analyzing ellipses in mathematical contexts.