Problem 213 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

An ellipse passes through the foci of the hyperbola, \(9 x^{2}-4 y^{2}=36\) and its major and minor axes lie along the transverse and conjugate axes of the hyperbola respectively. If the product of eccentricities of the two conics is \(\frac{1}{2}\), then which of the following points does not lie on the ellipse? |Online April \(\mathbf{1 0}\), 2015| (a) \(\left(\sqrt{\frac{13}{2}}, \sqrt{6}\right)\) (b) \(\left(\frac{\sqrt{39}}{2}, \sqrt{3}\right)\) (c) \(\left(\frac{1}{2} \sqrt{13}, \frac{\sqrt{3}}{2}\right)\) (d) \((\sqrt{13}, 0)\)

Step-by-Step Solution

Answer

Point (d) \((\sqrt{13}, 0)\) does not satisfy the ellipse equation.

Step 1: Identify the Hyperbola Parameters

The given equation of the hyperbola is \(9x^2 - 4y^2 = 36\), which can be rewritten as \(\frac{x^2}{4} - \frac{y^2}{9} = 1\). This is the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where \(a^2 = 4\) and \(b^2 = 9\). Hence, \(a = 2\) and \(b = 3\). The foci of the hyperbola are \((\pm\sqrt{a^2 + b^2}, 0) = (\pm\sqrt{13}, 0)\). The eccentricity of the hyperbola is \(e = \frac{\sqrt{a^2 + b^2}}{a} = \frac{\sqrt{13}}{2}\).

Step 2: Analyze the Ellipse Parameters

Given that the major axis of the ellipse aligns with the transverse axis of the hyperbola and the minor axis with the conjugate axis, we can describe the ellipse as \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) or \(\frac{x^2}{b^2} + \frac{y^2}{4} = 1\). Its eccentricity is \(e' = \sqrt{1 - \frac{4}{b^2}}\). We also know \(e \cdot e' = \frac{1}{2}\). Substituting \(e = \frac{\sqrt{13}}{2}\), we get \(e' = \frac{1}{\sqrt{13}}\).

Step 3: Solve for Ellipse’s Semi-Major Axis

From the ellipse's eccentricity equation, \(e'^2 = \frac{1}{13} = 1 - \frac{4}{b^2}\). Solving this equation, we find that \(\frac{4}{b^2} = \frac{12}{13}\), leading to \(b^2 = \frac{52}{13} = 4\). Thus, \(b^2 = 4 + a^2\), reaffirming \(a^2 = 4\) and thus \(b^2 = \frac{52}{13}\).

Step 4: Check Which Point Does Not Lie on the Ellipse

We now check the given points to see which do not satisfy the ellipse equation. Substitute \(x\) and \(y\) for each point into: \(\frac{x^2}{b^2} + \frac{y^2}{4} = 1\) with \(b^2 = \frac{52}{13}\) to find which does not hold true.

Key Concepts

Conic Sections

The study of conic sections involves the curves obtained from the intersection of a plane with a double-napped cone. These curves include circles, ellipses, parabolas, and hyperbolas. Understanding these is crucial as they frequently appear in math and physics challenges. Each shape arises depending on the angle and position at which a plane intersects the cone.

- **Circle**: A set of points equally distant from a center within a plane. It is the result of a perpendicular cut through the cone.
- **Ellipse**: This is an oval-shaped curve like an extended circle. It arises when the cut is inclined but does not pass through the base.
- **Parabola**: This U-shaped curve forms when the intersection is parallel to the cone's side.
- **Hyperbola**: Distinct with its two separate branches, this appears when the plane passes through both nappes of the cone.

Both ellipses and hyperbolas have real-world applications, such as in planetary orbits or satellite dish designs. Understanding conic sections covers recognizing these intersections and translating parameters into geometric insights, paving the way for analysis involving these pivotal shapes.

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being circular. For each type of conic, the eccentricity, denoted as \(e\), holds distinctive significance. It helps define the shape's roundness or openness:

- **Circle**: Here, \(e = 0\), as a circle is perfectly symmetrical around its center.
- **Ellipse**: For an ellipse, \(0 < e < 1\). The closer \(e\) is to 0, the rounder the ellipse becomes. It shows how stretched out the shape is.
- **Parabola**: With \(e = 1\), the parabola's arms stretch infinitely apart.
- **Hyperbola**: With \(e > 1\), hyperbolas have a more pronounced spatial expansion.

In the problem, the relationship of eccentricities provides a vital clue. The product of the elliptic eccentricity and hyperbolic eccentricity is given as \(\frac{1}{2}\), showing the correlation between both conics' shapes. This relation assists in determining the ellipse's parameters and ultimately solving the problem.

Standard Form Equation

Understanding the standard form equations for conic sections is crucial for solving related problems. These equations offer a structured way to identify and manipulate conics based on their algebraic representations.

For instance, in the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), \( a \) and \( b \) represent distances that define the hyperbola's axes shape and size. For ellipses, the standard form becomes \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), indicating proportions and orientation:

- **Ellipse**: Characterized by a plus sign, denoting an enclosed shape. Here, \(a\) and \(b\) help denote the semi-major and semi-minor axes lengths respectively.
- **Hyperbola**: Featuring a minus sign, this parameter configures the characteristic open double-curve.

In the provided exercise, converting equations to standard forms facilitates understanding how each conic's specific parameters relate geometrically and algebraically. Recognizing this, combined with eccentricity, allows for solving complex problems involving the positions and properties of these curves.